the use of simulation to expedite experimental ... · next, a system identification method to...
TRANSCRIPT
The Use of Simulation to Expedite Experimental Investigations of the Effect of High-Performance
Shock Absorbers
by
Christopher M. Boggs
Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in
Mechanical Engineering
Dr. Steve C. Southward, Co-Chairman Dr. Mehdi Ahmadian, Co-Chairman
Dr. John B. Ferris Dr. Saied Taheri Dr. Corina Sandu Dr. Leigh McCue
January 19, 2009 Danville, Virginia
keywords: vehicle dynamics, system identification, dynamic substructuring, modeling, simulation, 8-post rig, shock absorber, damper
Copyright 2009, Christopher M. Boggs
ii
The Use of Simulation to Expedite Experimental Investigations of the Effect of High-Performance Shock Absorbers
Christopher M. Boggs
Abstract
Successful race teams rely heavily on track testing to search for the ideal suspension setup. As more restrictions are placed on the amount of on-track testing by major racing sanctioning bodies, such as NASCAR, teams have increased their attention to alternate testing methods to augment their track data and better understand the dynamics of their racecars. One popular alternate to track testing is 8-post dynamic shaker rig testing. Eight-post rig testing gives the team a better understanding of the vehicle’s dynamics before they arrive at the race track, allowing them to use their limited track testing time more efficiently.
While 8-post rig testing certainly is an attractive option, an extensive test matrix is often required to find the best suspension setups. To take full advantage of 8-post rig tests, more efficient experimental methods are needed. Since investigating shock absorber selection is often the most time-consuming task, this study focuses on developing more efficient methods to select the best shock absorber setups.
This study develops a novel method that applies dynamic substructuring and system identification to generate a mathematical model that predicts the results of future tests as both command inputs and components are changed. This method is used to predict the results of 8-post rig tests as actuator commands and shock absorber forces are varied. The resulting model can then be coupled with shock absorber models to simulate how the vehicle response changes with shock absorber selection. This model can then be applied to experimental design.
First, a physically-motivated nonlinear dynamic shock absorber model is developed, suitable for quickly fitting experimental data and implementing in simulation studies. Next, a system identification method to identify a vehicle model using experimental data is developed. The vehicle model is then used to predict response trends as shock absorber selection is varied. Comparison of simulation and experimental results show that this model can be used to predict the response levels for 8-post rig tests and aid in streamlining 8-post rig testing experimental designs.
iii
Acknowledgements
I would first like to thank my advisors, Dr. Steve Southward and Dr. Mehdi Ahmadian,
for their guidance and support during this project and throughout my graduate studies in
the Mechanical Engineering Department at Virginia Tech. I would also like to thank Dr.
John Ferris, Dr. Saied Taheri, Dr. Corina Sandu, and Dr. Leigh McCue for their advice
and for serving on my graduate committee.
I would also like to thank the Virginia Institute for Performance Engineering and
Research (VIPER) for their support in providing excellent facilities and equipment for
my research along with ample laboratory time required to test this work. Special thanks
are due to Dr. Steve Southward and Bryan Pittman for serving as an 8-post rig operator
for all my rig testing. Thanks are also due to Mike Alex and Chandler Reubush in
providing me background on current trends in 7-post rig testing and analysis.
Finally, I would like to thank my family – my mother and father, Carolyn and Doug
Boggs, and my brother Gregory for their support during my studies at Virginia Tech.
iv
Contents
Abstract........................................................................................................................... ii
Acknowledgements......................................................................................................... ii
Acknowledgements........................................................................................................ iii
Contents ......................................................................................................................... iv
List of Figures............................................................................................................... vii
List of Tables ................................................................................................................ xii
Nomenclature............................................................................................................... xiii
CHAPTER 1 INTRODUCTION.............................................................................................. 1
1.1 Motivation................................................................................................................ 1
1.2 Objectives ................................................................................................................ 4
1.3 Approach.................................................................................................................. 4
1.4 Contribution ............................................................................................................. 6
1.5 Outline ..................................................................................................................... 7
CHAPTER 2 BACKGROUND ............................................................................................... 8
2.1 Analytical Framework ............................................................................................. 8
2.2 Suspension Design and the Role of the Shock Absorber....................................... 18
2.3 Shock Absorber Construction................................................................................ 20
2.4 Shock Absorber Modeling ..................................................................................... 24
v
2.5 8-Post Testing ........................................................................................................ 25
2.6 The Role of Experimental Data in Vehicle Modeling ........................................... 31
CHAPTER 3 SHOCK ABSORBER MODELING ................................................................... 34
3.1 Physical Motivation ............................................................................................... 34
3.2 Experimental Setup................................................................................................ 48
3.3 Shock Absorber Modeling ..................................................................................... 54
3.4 Model Validation ................................................................................................... 59
3.5 Summary................................................................................................................ 69
CHAPTER 4 QUARTER-CAR MODEL DEVELOPMENT .................................................... 71
4.1 Quarter-Car Model, Decoupled Analysis .............................................................. 72
4.2 Quarter-Car Model, Simulated Data...................................................................... 79
4.3 System Identification Methods .............................................................................. 81
4.4 System Identification on Simulated Data .............................................................. 88
4.5 System Identification for Quarter-Car Rig Data.................................................... 92
4.6 Issues with Extending to Full-Vehicle Modeling ................................................ 101
CHAPTER 5 FULL-VEHICLE MODELING AND IDENTIFICATION .................................. 103
5.1 Full-Vehicle Model.............................................................................................. 103
5.2 System Identification Methods ............................................................................ 118
5.3 System Identification Results .............................................................................. 128
5.4 Summary.............................................................................................................. 137
vi
CHAPTER 6 8-POST RIG TESTING AND IDENTIFICATION .............................................. 139
6.1 Experimental Setup.............................................................................................. 139
6.2 Linear Model Validity ......................................................................................... 152
6.3 8-Post Rig System Identification ......................................................................... 162
6.4 Model Validation ................................................................................................. 170
6.5 Summary.............................................................................................................. 178
CHAPTER 7 PREDICTING THE INFLUENCE OF SHOCK ABSORBER SETUP..................... 179
7.1 Shock Build Database.......................................................................................... 179
7.2 Simulated Shock Trends ...................................................................................... 182
7.3 Experimental Results ........................................................................................... 188
7.4 Summary.............................................................................................................. 190
CHAPTER 8 CONCLUSIONS ............................................................................................. 191
8.1 Summary and Conclusions .................................................................................. 191
8.2 Recommendations for Future Research............................................................... 194
REFERENCES................................................................................................................... 200
vii
List of Figures
Figure 1.1. Synergy Racing car on 8-post rig (photo by author, 2009) ......................................................... 2
Figure 1.2. Dynamic substructuring example................................................................................................ 4
Figure 2.1. Dynamic substructuring example................................................................................................ 9
Figure 2.2. Substructuring for car on track.................................................................................................. 11
Figure 2.3. (a) Vehicle dynamics present on 8-post rig, (b) Dynamics excited by 8-post rig .................... 11
Figure 2.4. Vehicle on 8-post rig and shock model substructuring ............................................................. 12
Figure 2.5. System identification flowchart ................................................................................................ 14
Figure 2.6. Penske 7300 shock absorber: (a) External view (photo by author, 2009), (b) Section view (adapted from [9], used with permission of Randy Lawrence, President, Penske Racing Shocks, 2009), (c) Diagram........................................................................................................................................................ 21
Figure 2.7. Construction of the main piston (adapted from [9] , used with permission of Randy Lawrence, President, Penske Racing Shocks, 2009)...................................................................................................... 22
Figure 2.8. Flows through the main piston during rebound: (a) Low speed bleed flow, (b) High speed valve flow (adapted from [9] , used with permission of Randy Lawrence, President, Penske Racing Shocks, 2009)............................................................................................................................................................. 23
Figure 3.1. Basic shock absorber construction ............................................................................................ 35
Figure 3.2. Equivalent fluid system for shock absorber .............................................................................. 36
Figure 3.3. Mechanical equivalents: (a) Damper in series with a spring, (b) Damper in series and in parallel with springs ..................................................................................................................................... 41
Figure 3.4. Sample shock dynamometer data and curve fit ......................................................................... 42
Figure 3.5. (a) Error in extrapolating a high-order polynomial fit, (b) Regions for polynomials in Polylinear model........................................................................................................................................... 43
Figure 3.6. LPF1 shock absorber model...................................................................................................... 44
Figure 3.7. The effect of a stiffening fluid on time constant: (a) Fluid stiffness, (b) Time constant .......... 45
Figure 3.8. Output weighting for LPNL2 model ......................................................................................... 46
Figure 3.9. Penske 7300 shock absorber on a Roehrig EMA (photo by author, 2009) ............................... 48
Figure 3.10. Force-velocity for the three shock builds tested: (a) Build 1, (b) Build 2, (c) Build 4 ........... 50
Figure 3.11. Segment of the random drivefile ............................................................................................. 52
viii
Figure 3.12. Static test drivefile................................................................................................................... 53
Figure 3.13. Bump test drivefile .................................................................................................................. 53
Figure 3.14. Polylinear model, force-velocity plot ...................................................................................... 55
Figure 3.15. Polylinear model, error vs. time .............................................................................................. 56
Figure 3.16. LPF1 model, force-velocity plot ............................................................................................. 57
Figure 3.17. LPF1 model, error vs. time...................................................................................................... 57
Figure 3.18. Random validation: (a) Build 1, (b) Build 2, (c) Build 4 ........................................................ 61
Figure 3.19. Force-velocity for build 2, 1 click: (a) 50 psi, (b) 150 psi....................................................... 62
Figure 3.20. Comparison of sine validation error versus model type for build 1, 14 clicks, 50 psi ........... 64
Figure 3.21. Force-velocity for sine tests: (a) low velocity, (b) high velocity............................................. 65
Figure 3.22. Static test time plots: (a) Drive profile, (b) Force ................................................................... 66
Figure 3.23. Static test force-displacement.................................................................................................. 67
Figure 3.24. Force vs. time for build 4 bump test: (a) Compression, (b) Rebound ..................................... 68
Figure 3.25. Force-velocity plot for build 4 bump test ................................................................................ 69
Figure 4.1. The quarter-car model ............................................................................................................... 73
Figure 4.2. Effect of damping on tire force ................................................................................................. 76
Figure 4.3. Effect of damping on ride height............................................................................................... 77
Figure 4.4. Effect of suspension stiffness on tire force................................................................................ 78
Figure 4.5. Effect of suspension stiffness on ride height............................................................................. 78
Figure 4.6. Coupling of vehicle and shock absorber models ....................................................................... 79
Figure 4.7. Simulink layout for quarter-car model ...................................................................................... 81
Figure 4.8. Diagram of complete system and the subsystem to identify ..................................................... 82
Figure 4.9. System ID experiments: (a) Linear shock for road and aero inputs, (b) Second shock for shock input.............................................................................................................................................................. 85
Figure 4.10. ID results for tire force............................................................................................................ 89
Figure 4.11. ID results ride height ............................................................................................................... 90
Figure 4.12. Simulink layout for simulation using the ID model ................................................................ 91
Figure 4.13. Relative amplitude spectra for drivefile .................................................................................. 92
ix
Figure 4.14. Comparison of tire force standard deviation ........................................................................... 92
Figure 4.15. Quarter-car rig at CVeSS (photo by author, 2009).................................................................. 94
Figure 4.16. Sensor mounting: (a) Accelerometer, (b) Shock potentiometer, (c) String potentiometer with integrated tachometer (photos by author, 2009) ........................................................................................... 95
Figure 4.17. Other components: (a) dSPACE AutoBox, (b) dSPACE Controldesk software, (c) MTS 407 controller (photos by author, 2009) .............................................................................................................. 96
Figure 4.18. Quarter-car rig FRF estimates ................................................................................................. 99
Figure 4.19. Coherence for FRF estimates .................................................................................................. 99
Figure 4.20. Comparison of simulated and measured time signals ........................................................... 100
Figure 4.21. Comparison of simulated versus measured response measures ............................................ 101
Figure 5.1. Chassis geometry and forces ................................................................................................... 105
Figure 5.2. Forces applied due to anti-rollbar............................................................................................ 108
Figure 5.3. Shock velocity frequency response ......................................................................................... 113
Figure 5.4. Tire force frequency response ................................................................................................. 114
Figure 5.5. Ride height frequency response .............................................................................................. 114
Figure 5.6. Coupling of vehicle and shock absorber models ..................................................................... 116
Figure 5.7. Full-vehicle model implemented in Simulink ......................................................................... 117
Figure 5.8. Diagram of system to identify................................................................................................. 118
Figure 5.9. Shock velocity FRF estimate: (a) FRF estimate, (b) Coherence ............................................. 129
Figure 5.10. Tire force FRF estimate: (a) FRF estimate, (b) Coherence ................................................... 130
Figure 5.11. Ride height FRF estimate: (a) FRF estimate, (b) Coherence................................................. 131
Figure 5.12. Comparison of ideal FRF, FRF estimate, and modified FRF estimate ................................. 133
Figure 5.13. Result of parametric identification ........................................................................................ 134
Figure 5.14. Simulink model ..................................................................................................................... 135
Figure 5.15. Simulation time trace comparison......................................................................................... 136
Figure 5.16. Simulation PSD comparison ................................................................................................. 137
Figure 5.17. Metric comparison ................................................................................................................ 137
Figure 6.1. 8-post rig: (a) NASCAR Cup car on rig, (b) Rig with pit cover removed, (c) Wheelloader and aeroloaders mounted under pit cover (photos by author, 2009) ................................................................. 140
x
Figure 6.2. Rig attachments to vehicle: (a) Wheel platen and wheel restraint, (b) Wheel platen load cells, (c) Aeroloader mounting and load cell (photos by author, 2009).............................................................. 141
Figure 6.3. Test control: (a) SmarTest controller, (b) FasTest PC interface (photos by author, 2009) .... 142
Figure 6.4. Test vehicle suspension: (a) Independent SLA front suspension, (b) Solid-axle trailing-arm rear suspension (photos by author, 2009) ................................................................................................... 143
Figure 6.5. Shock instrumentation: (a) Shock potentiometer, (b) Shock load cell (photos by author, 2009).................................................................................................................................................................... 144
Figure 6.6. Accelerometers: (a) Wheel and chassis accelerometers, (b) Cockpit accelerometers (photos by author, 2009) .............................................................................................................................................. 145
Figure 6.7. Chassis spring rating tests ....................................................................................................... 150
Figure 6.8. Shock builds: (a) Baseline, (b) Alternate ................................................................................ 151
Figure 6.9. Shock velocity due to wheelloader input: (a) Time response, (b) FRF magnitude, (c) Coherence................................................................................................................................................... 154
Figure 6.10. Shock velocity due to aeroloader input: (a) Time response, (b) FRF magnitude, (c) Coherence.................................................................................................................................................................... 155
Figure 6.11. Tire force due to wheelloader input: (a) Time response, (b) FRF magnitude, (c) Coherence.................................................................................................................................................................... 156
Figure 6.12. Tire force due to aeroloader input: (a) Time response, (b) FRF magnitude, (c) Coherence. 157
Figure 6.13. (a) Aeroloader and rear spring locations, (b) Simplified rear axle model .............................. 157
Figure 6.14. Ride height due to wheelloader input: (a) Time response, (b) FRF magnitude, (c) Coherence.................................................................................................................................................................... 158
Figure 6.15. Ride height due to aeroloader input: (a) Time response, (b) FRF magnitude, (c) Coherence.................................................................................................................................................................... 159
Figure 6.16. Shock velocity due to shock force input: (a) Change in time response, (b) FRF magnitude, (c) Coherence................................................................................................................................................... 160
Figure 6.17. Tire force due to shock force input: (a) Change in time response, (b) FRF magnitude, (c) Coherence................................................................................................................................................... 161
Figure 6.18. Ride height due to shock force input: (a) Changer in time response, (b) FRF magnitude, (c) Coherence................................................................................................................................................... 161
Figure 6.19. Total coherence for select outputs......................................................................................... 166
Figure 6.20. Shock velocity FRF estimate................................................................................................. 168
Figure 6.21. Tire force FRF estimate......................................................................................................... 168
Figure 6.22. Ride height FRF estimate...................................................................................................... 169
Figure 6.23. Hub acceleration FRF estimate ............................................................................................. 169
xi
Figure 6.24. Transfer function fits for shock velocity ............................................................................... 170
Figure 6.25. Simulink model ..................................................................................................................... 171
Figure 6.26. Baseline response for Richmond drivefile ............................................................................ 172
Figure 6.27. Shock setup RMS trending for Richmond drivefile .............................................................. 173
Figure 6.28. Shock setup RMS correlation for Richmond drivefile .......................................................... 174
Figure 6.29. Sine heave time results.......................................................................................................... 175
Figure 6.30. Sine heave testing RMS results ............................................................................................. 175
Figure 6.31. Bump testing results .............................................................................................................. 176
Figure 6.32. Static testing results............................................................................................................... 177
Figure 7.1. Front ride height simulation trending...................................................................................... 183
Figure 7.2. Front ride height simulation tradeoffs ..................................................................................... 184
Figure 7.3. Total tire force simulation trending......................................................................................... 185
Figure 7.4. Total tire force simulation tradeoffs........................................................................................ 186
Figure 7.5. Right front hub acceleration simulation trending.................................................................... 187
Figure 7.6. Right front hub acceleration simulation tradeoffs ................................................................... 187
Figure 7.7. Comparison of simulated and measured shock trending ......................................................... 189
xii
List of Tables
Table 2.1. Comparison of vehicle excitation and behavior on track and on 8-post rig................................ 28
Table 3.1. Shock configurations tested........................................................................................................ 50
Table 3.2. Definition of relative amplitude spectrum.................................................................................. 51
Table 3.3. Summary of sine wave tests ....................................................................................................... 53
Table 3.4. Summary of model fitting for build 1, 50 psi, 1 click................................................................. 59
Table 3.5. RMS error analysis, lb................................................................................................................ 60
Table 4.1. Baseline parameter values .......................................................................................................... 76
Table 4.2. Comparison of natural frequency and damping ratios................................................................ 90
Table 4.3. Relative amplitude spectrum for random drivefile ..................................................................... 96
Table 4.4. Sine wave tests ........................................................................................................................... 97
Table 4.5. Shock configurations tested........................................................................................................ 97
Table 5.1. Model parameters ..................................................................................................................... 112
Table 6.1. Summary of measured and calculated channels ....................................................................... 145
Table 6.2. Sine test frequencies and amplitudes........................................................................................ 147
Table 6.3. Random signal relative amplitude spectrum............................................................................. 148
Table 6.4. Importance of coherence and output energy for linear modeling ............................................. 153
Table 6.5. Zeroed input-output pairs ......................................................................................................... 166
Table 6.6. Number of active feedback paths ............................................................................................. 167
Table 7.1. Number of shock builds and simulation times.......................................................................... 180
Table 7.2. Shim sizes (adapted from [9], used with permission of Randy Lawrence, President, Penske Racing Shocks, 2009)................................................................................................................................. 181
Table 7.3. Shock build database ................................................................................................................ 182
Table 7.4. RMS level extremes ................................................................................................................. 183
Table 7.5. Shock setups for 8-post rig testing ........................................................................................... 188
xiii
Nomenclature
This section defines all major variables used in this document. While the nomenclature
section for each chapter lists all the variables used in that chapter for convenience,
consistent nomenclature across all chapters was used in most cases.
Chapter 2
t∆ Time increment
[ ]u k , [ ]y k Input and output at time index k
kU , kY Input and output time history up to time index k
( ), , ,LF RF LR RRM θ θ θ θ Input-output mapping for 8-post rig system parameterized by shock parameters
, , ,LF RF LR RRθ θ θ θ Shock parameterization
Θ Shock selection space
( ),K KJ U Y Objective function
( ),K KG U Y Response constraints
( )shock shockM θ Shock substructure mapping
( )car carM θ Car substructure mapping
Chapter 3
cQ , rQ Rate of change of volume for compression and rebound chambers
gQ Flow that displaces gas piston
pQ Flow through main piston
cQ , rQ Effective compressive flows for compression and rebound chambers
xiv
cp , rp Compression and rebound chamber pressure
cop Initial chamber pressure
ck , rk Stiffness of compression and rebound chamber compressibility
gk Gas chamber stiffness
'gk Stiffness of gas spring and compression chamber in series
C Flow resistance for main piston
cA , rA Piston area on compression and rebound sides
rodA Piston rod area
x , v Piston rod displacement and velocity
F Shock absorber force
incC , incK , oF Incompressible shock absorber damping, stiffness, and preload force
s Laplace variable
τ Shock model time constant
pk , sk , c Equivalent mechanical model parallel stiffness, series stiffness, and damping
sF , ( )f v Algebraic shock force
lagF LPF1 force
( )sFα LPNL2 weighting function
( )unscaledx t Unscaled time signal
iA , if , iφ Amplitude, frequency, and phase at ith frequency
Chapter 4
1m , 2m Sprung and unsprung masses
k , tk Suspension and tire stiffness
c Damping
xv
1w , 2w Sprung and unsprung mass displacement
Ry , AF , SF Inputs from road, aerodynamics, and shock
w Quarter-car degrees of freedom
M , C , K , L State-space mass, damping, stiffness, and input matrices
x State matrix
u Input matrix
y Output matrix
A , B , C , D Continuous-time state-space matrices
dA , dB , dC , dD Discrete-time state-space matrices
sx Suspension travel
RHx Ride height
springF , tF Spring and tire forces
g Gravitational constant
H Frequency response function (FRF)
s Laplace operator
T Sample time
z Unit-time advance operator
j Imaginary unit
ω Frequency, radians/second
f Frequency, Hz
polylinearF Polylinear force
2LPNLF LPNL2 force
τ Time constant
1H , 2H , 3H FRF matrices
1u , 2u , 3u Road, aerodynamic, and shock force inputs
ε Modeling error in frequency domain
Y , U Outputs and inputs in frequency domain for all data blocks
xvi
J Objective function
uuS , uyS Input spectrum and cross spectrum
shockF Shock force
c Linear damping
v Shock velocity
S Matrix to select shock velocity from output vector
1H , 2H FRF matrices with linear shock
1H , 2H Shock force FRF with linear shock
experiment 1y Output from baseline experiment
( )B s , ( )A s , nb , ma Transfer function numerator and denominator polynomials and their associated polynomial coefficients
( )tw k Weighting function
Chapter 5
Vehicle Parameters
t , a , b Track width and dstance from CG to front and rear axles
M , Jθ , Jφ Chassis inertial properties
LFtm , RF
tm , LRtm , RR
tm Wheel masses
Vehicle Position
1LFz , 1
RFz , 1LRz , 1
RRz Chassis vertical displacements at four corners
2LFz , 2
RFz , 2LRz , 2
RRz Wheel vertical displacements at four corners
z , θ , φ Chassis heave, roll, and pitch
1z Chassis vertical displacements vector
2z Wheel vertical displacements vector
Φ Chassis heave, roll, and pitch vector
xvii
T Transformation from chassis mode displacements to vertical displacements
Inputs
zF , Mθ , Mφ External forces applied to chassis
LFRz , RF
Rz , LRRz , RR
Rz Road input at four corners
Rz , extF , DF Road inputs, external chassis forces, and arbitrary suspension forces
7DOF Model Development
cM , tM Diagonal mass matrices for chassis and tires
LFF , RFF , LRF , RRF Suspension forces at four corners
LFtF , RF
tF , LRtF , RR
tF Tire forces at four corners
cF , tF Suspension and tire force vectors
LFk , RFk , LRk , RRk Suspension stiffness LFtk , RF
tk , LRtk , RR
tk Tire stiffnesses
sK , tK , PC 4x4 diagonal suspension stiffness, tire stiffness, and suspension damping matrices
PK 7x7 primary suspension and tire stiffness matrix
FRk , R
Rk Front and rear anti-rollbar stiffnesses
1K , rollK 4x7 and 7x7 anti-rollbar stiffness matrix
Structural and State-Space Equations
Z Degrees of freedom
M , sC , K , L Mass, damping, stiffness, and input matrices
u , y , x Input, output, and state vectors
A , B , C , D State-space matrices
Output Equations
shockv Shock absorber velocity
xviii
FRH , RRH Front and rear ride heights
1l , 2l , l Ride height geometry
FRF Estimation
rigu , shocksu Input vectors corresponding to actuators and shock forces
rigB , shocksB Input matrices corresponding to actuators and shock forces
rigD , shocksD Output matrices corresponding to actuators and shock forces
s Laplace operator
rigH , shocksH Frequency response due to actuator and shock force inputs
S Matrix to select shock velocity from output vector
Y , U Outputs and inputs in frequency domain for all data blocks
E Modeling error in the frequency domain
H Frequency response function (FRF)
J Objective function
uuS , uyS Input spectrum and cross spectrum
C Linear damping
AH FRF matrix with linear shocks
F∆ Arbitrary shock force in addition to linear shock force
1f∆ , 2f∆ , 3f∆ , 4f∆ Arbitrary shock force in addition to linear shock force
1shocksu , 2
shocksu , 3shocksu , 4
shocksu Shock force inputs
1c , 2c , 3c , 4c Linear damping coefficients
1sv , 2
sv , 3sv , 4
sv Shock velocities
rigH , shocksH FRF matrices with linear shocks and an additional arbitrary shock force
rigU Inputs for all blocks applied to baseline setup
'1c , 1c∆ Alternate left front shock damping coefficient and change
in damping coefficient from baseline
xix
[ ]0y k , [ ]0 kε Output and error for kth baseline data block
iy , Tiu , iε Output, input, and error for ith alternate experiement
rigE Error for all data blocks for baseline setup
rigi H
J Objective function for ith alternate experiment given the
FRF estimate for the baseline inputs
[ ]iy k , [ ]iu k Output and input for ith alternate experiment and kth data block
[ ]iy k∆ , [ ]if k∆ , [ ]i kε Change in output, change in shock force, and modeling error for ith alternate experiment and kth data block
iY∆ , iF∆ , iE Change in output, change in shock force, and modeling error for ith alternate experiment and all data blocks
ishockH Shock FRF for ith shock
f Frequency, Hz
e Euler’s number
j Imaginary unit
highH , lowH High and low frequency residual fit form
Transfer Function Fitting
( )B s , ( )A s , nb , ma Transfer function numerator and denominator polynomials and their associated polynomial coefficients
( )tw k Weighting function
Chapter 6
X , mX , xT Aeroloader displacements, aeroloader mode displacements, and the associated transformation
F , mF , fT Aeroloader forces, aeroloader mode forces, and the associated transformation
'b , 'L , 'a , ft , rt Aeroloader geometry about reference point along vehicle centerline
xx
y Output vector
rigu , shocksu Input vectors corresponding to actuators and shock forces
rigH , shocksH Frequency response due to actuator and shock force inputs
C Linear damping
shocksv Shock absorber velocity
F∆ Arbitrary shock force in addition to linear shock force
S Matrix to select shock velocity from output vector
rigH , shocksH FRF matrices with linear shocks and an additional arbitrary shock force
( ) [ ]iy k , [ ]rigu k Output and input for ith alternate experiment and kth data block
( ) [ ]iF k∆ , ( ) [ ]i kε Change in shock force and modeling error for ith alternate experiment and kth data block
Y , U Outputs and inputs in frequency domain for all data blocks
E Modeling error in the frequency domain
H Frequency response function (FRF)
ijh Logical matrix with zeroes if corresponding entry in shocksH was zeroed
f Frequency, Hz
aerow , wheelsw , shocksw Weighting function for aeroloader, wheelloader, and shock force transfer function fitting
aeroF , aeroX , zeroF Aeroloader force, displacement, and force at zero displacement
K Spring matrix, expressed in terms of aeroloader forces and displacements
mK Spring matrix, expressed in terms of aeroloader mode forces and displacements
Chapter 1 Introduction
The following sections describe the motivation for the proposed research, research
objectives, the approach, and the expected contributions.
1.1 Motivation
Successful race teams search for every possible advantage over the competition, no
matter how small, to reduce lap times and win the race. While track testing is ultimately
the most valid testing method available, the amount of time and resources track testing
takes is prohibitive when searching for the perfect setup. As more restrictions are placed
on the amount of on-track testing by major racing sanctioning bodies, such as NASCAR,
teams have increased their attention to alternate testing methods to augment their track
data and better understand the dynamics of their racecars.
One popular alternative to track testing that can be used to better understand the vehicle’s
dynamics is shaker rig testing, such as 7-post or 8-post rig testing, as shown in Figure 1.1.
Seven-post rig testing is widely used in Formula-One racing, but has become popular in
NASCAR only in the last few years [1, 2]. During a 7-post or 8-post rig test, the tires are
supported by 4 actuated wheel platforms, or wheelloaders, that simulate inputs from the
track surface. Three or four actuators, or aeroloaders, attach to the vehicle’s chassis to
provide forces which simulate the effect of inertial and aerodynamic forces present
during a track test. The vehicle is shaken on the rig to simulate how the car would
respond at a particular track or to characterize the vehicle response to more general
waveforms such as sine waves or broadband signals. In this study all tests will be
conducted on an 8-post rig, but all work is applicable to a 7-post rig, so we will refer to 8-
post testing unless specifically referring to 7-post testing.
2
Figure 1.1. Synergy Racing car on 8-post rig (photo by author, 2009)
Eight-post testing is an attractive option to race teams, and has become very popular in
the past several years. Unlike track testing, rig testing can be conducted as much as the
team wants. Also, the costs due to facility rental, travel, number of personnel, gas, tires,
etc. is much lower for an 8-post rig test. The test conditions are more repeatable during
an 8-post rig test than during a track test, which makes it easier to distinguish between the
effect of changing the setup and test variability.
While 8-post rig testing certainly has advantages, there are also downsides. Some of the
major challenges will be discussed in further detail in Chapter 2. One issue with 8-post
rig testing is that while it is more efficient than track testing, an extensive test matrix can
still take a long time. To take full advantage of 8-post rig tests, more efficient
experimental design methods are needed.
During an 8-post rig test, different vehicle adjustments are made to determine the effect
of each adjustment and to find ideal setups for different track conditions. Some of the
parameters that can be considered in a typical 8-post rig test are shown in Table 1.1.
3
Table 1.1. Potential 8-post rig testing adjustments
Of the 63 possible adjustments, investigating the 48 potential shock absorber adjustments
can easily be the most time-consuming. In this study, we will use the terms shock and
shock absorber interchangeably. This burden is further confounded by the fact that most
of those shock adjustments are internal to the shock, so there needs to be a shock built for
each adjustment. This motivates us to determine efficient methods to vary the shock
parameters, since we have the most to gain by reducing this effort.
Clearly, the shock adjustments can add the most complexity to any 8-post rig test
experimental design. If we could find a way to characterize the effect of shock
adjustments, and use that information to run a shaker rig test more efficiently, then we
could significantly reduce the complexity of our tests.
While running a large test matrix will give the best results, it is often impractical. Most
experimental designs in 8-post testing are done in a guess-and-check manner. This
consists of the engineers guessing a list of potential setups, testing them, and checking the
test data. After checking the data, they may generate further lists. This method clearly
has problems. First, only information on those particular setups is gained, not more
general sensitivity information that could be used to tune shocks. Second, potential gains
can be missed if the setup is not part of the guessed list. Third, if the guesses are poor,
there is often no indication of what to try next. Clearly, this method is highly dependent
on the engineer’s ability to guess setups.
Most teams also test their shocks on a shock dynamometer. Since shocks can be the most
challenging components to tune during a rig test, knowing how the shock performs at a
component level before running shaker rig tests could be very useful. While finding a
practical method to apply shock data to 8-post rig testing remains a major challenge, the
potential improvement in testing efficiency could be significant.
12 shock settings per corner x4 = 481 spring rate and 1 preload per corner x4 = 81 rollbar rate per axle x2 = 21 trackbar height x1 = 11 tire pressure per tire x4 = 4
63 parameters
4
1.2 Objectives
The overall objective of this work is to develop methods to use race shock data from
shock dynamometer tests to improve shaker rig testing. The detailed objectives of this
study are to
1. Characterize the performance of race shocks,
2. Develop a validated mathematical model that uses prior 8-post rig test data to
predict the influence of shock absorber selection in future 8-post rig tests, and
3. Apply the model to improve the efficiency of 8-post tests.
1.3 Approach
This study develops a novel method that applies system identification for dynamic
substructuring to generate a mathematical model that predicts the results of future tests as
both command inputs and components are changed. Dynamic substructuring represents
the behavior of a complex system as an assembly of simpler substructures and their
interactions, as shown in Figure 1.2. This modeling technique allows us to investigate
the effect of changing various system components by swapping substructure models that
represent the system components. This study will apply system identification techniques
to identify mathematical models for all substructures from experimental data.
Substructure 1
Substructure 2
Substructure 3
System
Inputs Outputs
Interactions
Substructure 1
Substructure 1
Substructure 2
Substructure 2
Substructure 3
Substructure 3
System
Inputs Outputs
Interactions
Figure 1.2. Dynamic substructuring example
5
This method is used to predict the results of 8-post rig tests as actuator commands and
shock absorber forces are varied. The resulting model can then be coupled with shock
absorber models to simulate how the vehicle response changes with shock absorber
selection. This model can then be applied to experimental design.
The detailed steps that will be taken to achieve the objectives are:
Shock Modeling
1. Develop characterizations of race shock performance using a shock
dynamometer,
Quarter-Car Study
2. Develop a method to identify a vehicle model using simulated quarter-car rig
data,
3. Run experiment on quarter-car rig and apply identification method,
8-Post Rig Study
4. Expand method to full scale using simulated 8-post rig data,
5. Apply method to 8-post rig data and validate model accuracy, and
Apply Method
6. Use model to predict vehicle performance for different setups and help select
setups, and
7. Conduct rig tests for different shock setups to validate the model’s ability to
predict vehicle performance and select setups.
Step 1 involves developing shock absorber models using shock dynamometer data. First,
a series of shock dynamometer tests were run on different shock builds, which are used
for both model fitting and validation purposes. Candidate shock models are developed,
which are then fitted with the experimental data. Additional dynamometer data is used to
validate the model performance and see where the model performs well.
6
To make it easier to develop the vehicle system identification method, it was done first at
a quarter-car scale in Steps 2 and 3. To make sure the identification method can extract
the vehicle model accurately, the method is first applied to simulated data in Step 2
generated by a known quarter-car model. Once the system identification method has
been proven on simulated data, it is then applied to test data collected from a quarter-car
rig in Step 3 to verify that the method will work on real data.
The method is expanded to full 8-post rig scale in Steps 4 and 5. Similar to the quarter-
car study, the method is first developed using simulated 8-post rig data to work out the
bugs before it is applied to actual rig data.
Once the 8-post rig vehicle model is created, Step 6 applies the model to help select
shock setups and identify sensitivity information. The results from Step 6 are validated
in Step 7 by running a series of rig tests for some of the same setups considered in Step 6.
1.4 Contribution
As discussed above, the contribution of this research to both the racing industry and the
larger engineering community is significant. The expected contributions of this work to
the racing industry include
1. Providing more accurate empirical shock absorber models that can be used for
simulation studies,
2. Providing a method to identify an empirical model of the influence of
component selection during 8-post rig testing, and
3. Improving the efficiency of 8-post test sessions by using model to identify
trends and help shock selection.
This work may also have a larger impact on the engineering community since
1. The method may be extended to include additional components,
7
2. The models can be used to suggest targets for shock design or be used as a
part of a larger vehicle simulation, and
3. The models and methods may be valid for different classes of vehicles or
other dynamic systems.
1.5 Outline
This study begins by developing shock absorber models, proceeds by developing the
vehicle model identification method at quarter-car and full-car scales, and concludes by
applying the identified model to predict trends in the vehicle response as shock setups are
changed. Chapter 2 covers relevant background information. Chapter 3 covers the shock
absorber modeling. Chapter 4 covers the quarter-car study. Chapter 5 develops the
method at full-vehicle scale using simulated data from a 7 degree of freedom model.
Chapter 6 applies the method to actual 8-post rig data and validates the model with
additional data sets. Chapter 7 applies the model identified in Chapter 6 to determine
which setups will have a high or low RMS response level, which is then verified for
several setups on the 8-post rig. Chapter 8 provides summary, conclusions, and
recommendations for future work.
8
Chapter 2 Background
This chapter presents background information that the reader may find useful while
reading this document. The chapter begins by casting this research in a larger analytical
framework suitable for a larger class of engineering problems. It continues by
highlighting several design requirements of a racing suspension and the role of the shock
absorber. Next, the construction of the shock absorber used in this study is described.
Then, prior work in shock absorber modeling is covered. Next, a summary of the
development of 8-post technology and the published work in that area is presented.
Finally, the role of experimental data in developing quality vehicle models is discussed.
2.1 Analytical Framework
While this research focuses on the application of shock selection during an 8-post rig test,
the approach and methods used in this research are applicable for a larger class of
engineering problems. To highlight how this research is applicable for a larger class of
problems, this section presents this research within a more general analytical framework.
First, the concept of dynamic substructuring is presented as a convenient approach for
manipulating complex models. After presenting the concept of dynamic substructuring,
the concept will then be applied to the problem of shock selection during 8-post rig
testing. The 8-post rig test will be shown to be a test of a substructure of the more
complex vehicle system. The vehicle during the 8-post rig test may be further
substructured into the components of interest. In this study our focus is shock absorbers,
so the vehicle on the 8-post rig will be substructured into 4 substructures for the shock
absorbers and one main substructure representing the vehicle and any remaining vehicle
components. Once the vehicle substructuring has been established, this framework can
then be used to state the shock absorber optimization problem and the 8-post rig
9
substructure system identification problem. At this point, we will formally state the
problem addressed in this research.
2.1.1 Dynamic Substructuring
Dynamic substructuring is a strategy to model a complex system as an assembly of
substructures and interactions between the substructures. An example of substructuring a
system into three substructures is shown in Figure 2.1. Substructuring involves dividing
a larger structure into substructures, which are chosen to best suit the application. Each
substructure is treated as a separate system, with inputs and outputs that are used to
couple the substructures, accept the system inputs, and generate the system outputs. In
mechanical systems, these interactions typically consist of a compatibility constraint on
displacement and an equilibrium constraint on force.
Substructure 1
Substructure 2
Substructure 3
System
Inputs Outputs
Interactions
Substructure 1
Substructure 1
Substructure 2
Substructure 2
Substructure 3
Substructure 3
System
Inputs Outputs
Interactions
Figure 2.1. Dynamic substructuring example
Since substructuring allows the user to change model components as a substructure, the
technique can be thought of as a compromise between black-box modeling and physical
modeling.
Some of the potential advantages of substructuring include
1. Using a combination of empirical and analytical models;
2. Allowing work to be done on a simpler system, which can be useful for
a. Model Derivation,
10
b. Experiments,
c. Analysis,
d. Simulation, and
e. System Identification;
3. Exploring the effect of changing components and geometry by changing
substructures and topology;
4. Increasing computational efficiency; and
5. Predicting the internal response, such as forces applied to a component.
2.1.2 Vehicle Substructuring
A car on the road may be substructured by drawing a control surface around the chassis,
cutting through the shocks, springs, and control arms. This control surface will also
intersect the brakes, throttle, and steering, which are controlled by the driver. These
components act to couple the chassis to the wheels. At each cut, the interaction is
represented by geometric compatibility and force equilibrium. The tire is mounted on the
wheel, and the tire interacts with the road surface. Also, aerodynamic forces are
generated due to the chassis velocity, height from the road surface, and orientation. This
substructuring is shown in Figure 2.2. The vehicle on the 8-post rig is consistent with the
subsystems enclosed by the dashed line.
On the 8-post rig, the forces applied are primarily in the vertical direction. While the
lateral and longitudinal tire grip on the low friction wheelloader platens will apply small
lateral and longitudinal forces to the vehicle, these are often not large enough to create
significant motion in the lateral, longitudinal, and yaw degrees of freedom. These
degrees of freedom, which may be considered the yaw-plane dynamics, are present on the
rig but are not significantly excited by the rig. On the track, these dynamics are excited
by the lateral and longitudinal tire grip and the inertial forces due to the vehicle’s
trajectory on the track. When on the track, the yaw-plane dynamics couple with the
remaining dynamics, which we will call the vertical dynamics. This coupling can be
represented as shown in Figure 2.3.
11
Aero
Chassis
Brakes Throttle SteeringControl Arms
Wheel
Tire
Shocks Springs
Road
Vehicle on 8-Post
Aero
Chassis
Brakes Throttle SteeringControl Arms
Wheel
Tire
Shocks Springs
Road
Aero
Chassis
Brakes Throttle SteeringControl Arms
Wheel
Tire
Shocks Springs
Road
Vehicle on 8-Post
Figure 2.2. Substructuring for car on track
Yaw-Plane Dynamics
VerticalDynamics
VerticalDynamics
(a) (b)
Yaw-Plane Dynamics
VerticalDynamics
Yaw-Plane Dynamics
VerticalDynamics
VerticalDynamics
(a) (b)
Figure 2.3. (a) Vehicle dynamics present on 8-post rig,
(b) Dynamics excited by 8-post rig
Since the 8-post rig test only excites a substructure of the complete vehicle system that
we wish to optimize, optimization of the vehicle performance during an 8-post rig test
becomes a challenging task. There are several methods that can be applied to apply 8-
post rig data to improve performance on the race track, which include
1. Correlation: Neglecting the coupling and tuning for metrics on the rig that have
shown correlation with good track performance in previous studies,
12
2. Hardware-in-the-Loop: Including a model of the dynamics not excited on the 8-
post rig and modifying the actuator commands to reflect how the coupling would
affect the vertical dynamics, and
3. Identification and Synthesis: Using the data collected from an 8-post rig test to
identify a model of the vertical dynamics to be synthesized with other components
for vehicle simulations.
In this study, we will apply method 1. The model that this work produces could also be
integrated into a larger vehicle model as part of method 3. Method 2 requires actually
modifying the actuator commands during the 8-post rig test and is not used here.
2.1.3 8-Post Rig Substructuring
Since this study focuses on shock absorber selection on the 8-post rig, we substructure
the 8-post rig model into a car model and 4 shock absorber models, as shown in Figure
2.4. Each model block consists of a model structure ( )M i , which is parameterized by
the parameter vector θ ∈Θ .
( )car carM θ
( )shock LFM θ ( )shock RFM θ ( )shock LRM θ ( )shock LRM θ
Actuators Sensors
Vehicle on 8-Post Rig
( )car carM θ
( )shock LFM θ ( )shock RFM θ ( )shock LRM θ ( )shock LRM θ
( )car carM θ
( )shock LFM θ ( )shock RFM θ ( )shock LRM θ ( )shock LRM θ
Actuators Sensors
Vehicle on 8-Post Rig
Figure 2.4. Vehicle on 8-post rig and shock model substructuring
Using this framework, we can define the problems of shock optimization and system
identification that will be addressed in this research.
13
2.1.4 Optimization Problem
For a given car, we wish to select the best shock absorbers out of a set of candidate shock
absorbers. The problem of shock absorber selection may be defined more formally as an
optimization problem. The quality of a particular shock selection will be determined by a
series of tests that will be conducted and an objective measure that will be calculated
from the recorded data. There may also be secondary measures that must not exceed
certain levels for an acceptable setup. This can be stated formally as an optimization
problem.
Define k to be the sample index corresponding to time k∆t, where ∆t is the sampling
time. We then define u[k] and y[k] to be the vector inputs and outputs recorded at
sample index k. Also define
[ ] [ ] [ ][ ] [ ] [ ]1 2
1 2
k
k
U u u u k
Y y y y k
= =
(2.1)
which is all the sampled signals up to time index k. Defining the test to be K samples
long, the matrix KU defines the drivefiles used for rig testing. The dynamics of the 8-
post rig can then be described by the nonlinear mapping
( ), , , : K KLF RF LR RRM U Yθ θ θ θ → (2.2)
where the parameters iθ define the shock selection, which may be selected from the
shock selection space Θ . The optimization problem of selecting the best shock
absorbers can then be stated
Minimize ( ),K KJ U Y
subject to
( ), , , : K KLF RF LR RRM U Yθ θ θ θ → (2.3)
, , ,LF RF LR RRθ θ θ θ ∈Θ
( ), 0K KG U Y ≤
14
where J represents the objective function and G represents constraints on secondary
response measures. This problem can be solved using gradient-based nonlinear
programming methods if the candidate shock model parameter space is continuous and
( )J i and ( )M i are differentiable. If the candidate shock model parameters are discrete,
integer programming methods must be used.
2.1.5 System Identification Problem
To perform the optimization problem shown in Equation (2.3) in simulation or
analytically, the dynamics of the vehicle on the 8-post rig and the influence of shock
selection must be known. A convenient method for determining the dynamics of the
system is to perform system identification on the substructures shown in Figure 2.4.
The generic system identification process is shown in Figure 2.5. The process is initiated
by defining a target system, which we wish to describe using a mathematical model. A
series of tests are performed on the target system, where the target system input and
output data is recorded. The experimental data is used both to perform system
identification and to assess the model quality. System identification applies the
experimental data to generate a mathematical model of the system. Once the model is
identified, the response predicted by the model is compared to the response measured
from the target system. If this comparison shows that the identified model is not
sufficient for the particular application, the testing or system identification processes
must be adjusted until an acceptable model is produced.
Target System Testing System
IdentificationAssess
Model QualityModel
Acceptable?ApplyModel
yes
no
Modify Testing or ID Methods
Target System Testing System
IdentificationAssess
Model QualityModel
Acceptable?ApplyModel
yes
no
Modify Testing or ID Methods
Figure 2.5. System identification flowchart
This study will first develop the testing and system identification methods in simulation
before performing laboratory testing. To develop the methods in simulation, a known
15
mathematical model is used for the target system. Simulation results are used as
simulated data sets, which can be used for system identification and assessing model
quality. By eliminating modeling error due to signal noise, system nonlinearity, and
repeatability, the simulation study will highlight weaknesses in the testing and system
identification processes.
To perform system identification, three things must be defined
1. A model structure ( )M i ,
2. A measure of model quality, and
3. A method of selecting model parameters given sufficient experimental data.
Another requirement is that the excitation signals must be sufficiently rich (sufficient
richness) and the model structure must be such that the true model parameters are a
unique optimal solution in terms of model quality (identifiability). This restriction
requires linearly independent excitation for each input and that the model structure is not
overly complex.
In this study, we will consider models for the vehicle and the shock absorbers. The
model for the shock will be
( ) : K Kshock shock shock shockM U Yθ → (2.4)
where KshockU and K
shockY are the input and output histories for the shock substructure.
Multiple shock absorbers will be tested at the component level on a shock dynamometer
to provide the data required for shock system identification. Since multiple shocks will
be tested and modeled, there will be a parameter vector shockθ for each shock tested. This
means that the space of candidate shock models, Θ , will consist of a finite number of
points. Since Θ is not continuous, integer programming methods will be used to solve
the optimization problem in Equation (2.3).
The model for the vehicle will be
16
( ) : K Kcar car car carM U Yθ → (2.5)
Where KcarU and K
carY are the input and output histories for the car substructure. Since we
will only test with one vehicle in this study, there will be only one parameter vector carθ .
One conventional testing method for collecting the data needed for system identification
would involve applying independent external forces to each input of the car substructure.
This research will focus on identifying the car substructure model in Equation (2.5) with
only data from complete 8-post structure testing and shock substructure testing.
2.1.6 Problem Statement
The main question that we attempt to answer in this research is:
Can the time required to find desirable shock absorber setups on the 8-post rig be
reduced using dynamic substructuring for system identification and optimization?
To perform this, we will need
1. Shock absorber models,
2. A model of the vehicle on the 8-post rig and how it interacts with the shock
absorbers, and
3. An objective function to rate the quality of different shock setups.
For the shock absorber substructure models, we will choose a model structure which is
motivated by the dominant velocity dependence of the device and the secondary
influence of fluid compressibility to add phase lag.
For the vehicle substructure model, we will use a linear model structure. The assumption
of linearity may prove to be insufficient to capture some of the dynamics, but it should
predict general trends that will help in shock selection. These issues will be discussed in
more detail later in this chapter.
The choice of a successful objective function remains a challenge. As discussed above,
the 8-post rig test can be considered a test of a substructure of the complete vehicle on the
17
racetrack. Rigorous analytical or simulation methods would require modeling of the
entire vehicle, including the tire/track interface, aerodynamics, braking, powertrain, and
steering. These models would either be used in a hardware-in-the-loop sense or to be
synthesized with a model of the vertical dynamics identified from the rig. Developing
and validating this level of modeling requires a large database of experimental data,
which is not available at this time and is beyond the scope of this research.
The most common approach in the 8-post testing industry is to use statistics, or response
measures, which have shown to be consistent with favorable track performance. The
selection of the proper response measures to analyze is typically based on the experience
of the engineers, not a formal correlation analysis. These response measures are typically
weighted combinations of the RMS or MS response of the tire force, ride height,
suspension deflection, and other measured or calculated signals in the form
( )1
RMSI
i ii
J yρ=
=∑ (2.6)
where iρ is the weighting factor for the RMS of signal iy . The signal may be bandpass
filtered to the frequency range of interest, and may include an entire test run or focus on a
critical segment of the test.
The models should be chosen by the user to sufficiently characterize the desired
dynamics for the application and to be evaluated quickly for efficient simulations. To be
considered successful, the resulting model must be fast enough to be run many times in a
day to provide suggestions for further 8-post testing.
In practice, the user would also need to select their objective function based on their best
knowledge of what is most appropriate for their application. Since we do not have a
validated objective function to suggest to users and this method could be used for a range
of applications, our metric of success will be the ability to predict how the RMS levels
shown in Equation (2.6) vary when changing shock setups. If we can successfully
identify trends in these RMS measures, we will consider this method a success.
18
2.1.7 Dynamic Substructuring Research
Most research in system identification in dynamic substructuring focuses on linear
second-order modal models [3]. In these studies, sensors must be placed on the structure
to identify the modes of interest and viscous or structural damping assumptions were
often made.
More recent work has applied more generic input/output model structures to
substructuring [4, 5]. The presented method identifies models of each substructure
independently, after which the substructure models are synthesized to create the structural
model. This method requires testing of all structures independently and also requires
excitation and sensors at all interface degrees of freedom. One method to provide
excitation at a boundary degree of freedom is to perform tests with multiple point masses
[4].
In 2008, a study was published that performs identification using a mixture of structure
and substructure data [6]. This is particularly useful when one of the structures is
difficult to test independently or behaves significantly different as part of the structure.
This method requires inputs to both substructures and the interface degrees of freedom.
2.2 Suspension Design and the Role of the Shock Absorber
The ability of the tire to maintain contact with the race track over bumps and around
corners is critical for achieving the best performance of a race car [7]. The suspension
determines the coupling between the chassis and the wheel, which determines how the
work done from external forces from the road and air propagate through the vehicle as
kinetic energy, potential energy, and dissipated energy. The proper design and tuning of
the suspension can mean the difference between the winning team and one that does not
even qualify for the race.
Some objectives for high-performance suspension design may include:
1. Providing the desired tire normal force distribution between the four tires,
19
2. Minimizing tire normal force variations due to disturbances,
3. Providing the desired body ride height and attitude,
4. Minimizing body ride height and attitude changes due to disturbances,
5. Controlling dynamic load transfer distribution and load transfer rate during
transient maneuvers,
6. Providing sufficient sensory feedback to the driver to allow for driving
decisions, and
7. Maintaining acceptable vehicle stability to allow for disturbances.
Unlike passenger car design, driver comfort is not typically a primary concern.
Tire traction or grip analysis is often simplified by breaking the total grip into
components of mechanical grip and aero grip [8]. Mechanical grip results from the
application of inertial forces and road disturbances to the vehicle, while aero grip is
caused by high-speed aerodynamic downforce. The best setup for a particular track will
be a tradeoff between the best mechanical grip and the best aero grip, which will depend
on the importance of aerodynamics on that particular race track. For example, at a half-
mile short track race, mechanical grip is often critical, while at a 2.5 mile superspeedway
track, it is often possible to sacrifice mechanical grip for improved aero grip.
To optimize mechanical grip, the tire normal forces must be distributed ideally for the
current track maneuver (braking, cornering, accelerating, etc.) and the normal force
variation must be minimal. The ideal normal force distribution will allow the four tires to
achieve the best lateral and longitudinal force balance for the given maneuver.
Minimal normal force variation will allow the driver to drive at the absolute limit of the
vehicle’s performance envelope. If there is more normal force variation, it will disturb
the available grip on the four tires. This will then disturb the vehicle’s lateral
acceleration, longitudinal acceleration, and yaw rate on the racetrack and the driver will
need to adjust by steering, changing the vehicle’s line, or slowing the vehicle. This
alteration from the ideal racing line can make lap times slower. Often disturbances occur
20
too fast for the driver to correct, so they will either need to adjust their driving style to
anticipate the disturbances or they may lose control. This anticipation often requires sub-
limit driving, which will also make lap times slower.
To achieve optimal aero grip, the chassis must maintain an ideal ride height and attitude
to provide the best total aerodynamic downforce for total grip and the desired downforce
distribution to the front and rear axles for balance. Any disturbance from this ideal
position will result in a change in downforce, which will cause a variation in the tire
normal force.
The suspension also controls dynamic load transfer during straight-line braking, turn
entry, turn exit, and straight line acceleration. A proper dynamic load transfer
distribution and load transfer rate are also important for good handling.
Two of the main components that can be tuned in a suspension are the springs and the
shocks. Quite often, however, the selection of spring rates, anti-rollbar stiffnesses, and
roll center heights is constrained by the desired steady-state load transfer distribution and
the desired DC chassis position at top speeds [7]. This means that the shock absorbers
often have the most freedom to be tuned for mechanical disturbances, aero disturbances,
and dynamic load transfer.
The shock absorber converts energy stored in the springs and the work done by external
forces into heat instead of kinetic energy of the chassis and wheel. This is done by
forcing a fluid through a restriction with a flow rate that is a function of the velocity
across the suspension. Since the shock absorber is a dissipative element, it is particularly
well-suited for controlling resonance, dynamic balance, and load transfer rate.
2.3 Shock Absorber Construction
This study focuses on the Penske 7300 shock absorber, shown in Figure 2.6. This shock
is commonly used in NASCAR Nextel Cup and other similar racing series. This shock,
21
like many other racing shocks, is designed to be highly adjustable, and consists of several
components, described below.
(a)
Gas Chamber
Rebound Chamber
Piston
Gas Piston
External Adjuster
Head Valve (fixed)Head Valve Chamber
Compression Chamber
Gas Chamber
Rebound Chamber
Piston
Gas Piston
External Adjuster
Head Valve (fixed)Head Valve Chamber
Compression Chamber
(b)
Gas Piston
Compression Chamber
Head Valve (fixed)
Head Valve Chamber
Piston
Rebound ChamberOil
Nitrogen
Oil
Oil
Shock in Compression
Gas Chamber
Compression
Gas Piston
Compression Chamber
Head Valve (fixed)
Head Valve Chamber
Piston
Rebound ChamberOil
Nitrogen
Oil
Oil
Shock in Compression
Gas Chamber
Compression
(c)
Figure 2.6. Penske 7300 shock absorber: (a) External view (photo by author, 2009),
(b) Section view (adapted from [9], used with permission of Randy Lawrence,
President, Penske Racing Shocks, 2009), (c) Diagram
The shock is mounted on the car using two spherical joints, which allow rotational
motion and prevent the application of side loads to the shock, which would degrade
performance. On the top of the shock is a gas chamber, which is separated from the
hydraulic fluid by a floating gas piston. The main purpose of the gas chamber is to
compensate for the volume of the piston rod as it enters the shock body. The gas
chamber is pressurized with nitrogen gas using a Schrader valve on the top of the shock.
This pressurization preloads the hydraulic fluid, reducing the effects of fluid
compressibility and the potential for fluid cavitation due to pressure drops across the
piston.
Below the gas chamber is the head valve, which separates the head valve chamber from
the compression chamber. The head valve is threaded into the shock body, and can be
removed when rebuilding the shock. The head valve is designed to accept different bleed
jets and shim stack configurations, which can be used to fine-tune low speed and high
22
speed shock behavior, respectively. The main purpose of the head valve is to provide a
higher pressure in the compression chamber during the compression stroke, forcing fluid
into the rebound chamber to avoid cavitation.
Below the head valve is the compression chamber, which is separated from the rebound
chamber by the main piston, shown in Figure 2.7. The main piston provides most of the
energy dissipation in the shock, since it sees more flow than the head valve. This makes
it the main factor that determines the shock’s force-velocity characteristic. The
components of the main piston assembly include the piston, compression shim stack,
rebound shim stack, bleed jet, and needle valve. Large-scale shock curve adjustments
can be made by selecting different piston types, which changes the valving. The bleed jet
threads into the shaft, and can be exchanged to adjust low speed damping. The needle
valve allows for the bleed area to be adjusted by changing the needle’s position, which is
controlled by an external knob on the bottom of the shaft. This knob is indexed by
clicking the adjuster. The click setting can range from fully open or 1 click, to fully
closed or 30 clicks. The compression and rebound shim stack stiffness and preload can
be adjusted to control high speed damping.
Bleed Jet
Rebound Shim Stack
Compression Shim Stack
Needle Valve
Bleed Jet
Rebound Shim Stack
Compression Shim Stack
Needle Valve
Figure 2.7. Construction of the main piston (adapted from [9] , used with
permission of Randy Lawrence, President, Penske Racing Shocks, 2009)
Now that the shock components have been introduced, shock function will be described.
When the shock is extending or in rebound, the piston-shaft assembly moves downward
relative to the shock body. This reduces the volume of the rebound chamber, forcing
23
fluid from the rebound chamber to flow into the compression chamber through the main
piston. The fluid has two main paths through the main piston, as shown in Figure 2.8. At
low speeds, the fluid will flow through the bleed. As speed increases, the pressure drops
across the piston. When the pressure drop exceeds the preload in the rebound shims, the
rebound shim stack will open, causing valve flow.
(a)
(b)
Figure 2.8. Flows through the main piston during rebound:
(a) Low speed bleed flow, (b) High speed valve flow (adapted from [9] , used with
permission of Randy Lawrence, President, Penske Racing Shocks, 2009)
Also as the piston-shaft assembly moves downward, the combined volume of the
compression and rebound chambers increases due to the removal of the rod. This total
volume change must be replaced by fluid from the head valve chamber. Like the main
piston, there are also bleed and valve flow for the head valve. The loss of shock oil
volume in the head valve chamber is then compensated for by a downward motion of the
gas piston and expansion of the gas charge in the gas chamber.
The flow across the main piston bleed and valves causes a pressure drop in the fluid.
This pressure drop creates a net force on the piston, which must then be reacted by the
rod. This is the main source of the shock force, with smaller contributions due to seal
drag, bearing drag, and piston-shaft assembly mass.
24
2.4 Shock Absorber Modeling
Shocks are complex nonlinear devices whose internal dynamics have been modeled for a
variety of applications. Lang developed a comprehensive physical shock model in 1977
[10]. This model included approximately 80 parameters, and is not practical for multiple
parametric studies or simulations. Reybrouck created a model with only 20 parameters in
1994 that captured many important shock dynamics [11]. This model relied on semi-
empirical coefficients, and did not consider the effect of internal modifications on shock
performance. Talbott developed a model of an Öhlins NASCAR Cup shock in 2002 that
considered the effect of shock tuning parameters, such as shim stack stiffness, shim stack
preload, and bleed orifice area on shock performance [12]. Emmons extended Talbott’s
model to include the head valve used in current Penske NASCAR Cup shocks [13].
The physical models listed above are useful for predicting the effect of changing shock
internals prior to assembling and testing a particular build. However, once a particular
shock has been run through a series of tests on the dynamometer, a more efficient
empirical model can be made by neglecting internal dynamics and only fitting the input-
output characteristics. Compared to physical models, empirical models can be more
accurate and can be evaluated faster, which is better for simulation purposes.
Empirical dynamic models for shock absorbers can often be loosely classified into linear,
shape function, and universal approximators.
Linear models use linear dynamic systems, such as spring-damper combinations and
transfer functions, which are useful for analysis and fast for simulation, but are often very
inaccurate [14-17].
Shape function models use prior knowledge of the shock’s force-velocity characteristic to
specify a shape function, whose parameters reflect the shape of different parts of the
curve. These algebraic shape function methods provide decent approximations and some
physical insight, but often rely on transcendental functions which take longer to fit and
evaluate, which is less suitable for simulation [18, 19].
25
Universal approximator models apply a family of functions which have the property of
being able to approximate any well-behaved function to an arbitrary degree of accuracy.
Universal approximators include neural networks, radial basis functions, polynomials,
splines, and wavelets. While the universal approximation models theoretically offer great
potential in dynamic modeling, the generic model structure often requires a much higher
order model compared to an ad-hoc method that takes advantage of knowledge about the
system [19-22]. These high-order models cause slower computation speed and longer
time to fit the model. Another issue with these models is they have limited ability to
predict performance outside the range of the fit data, which may introduce large errors
[20].
One important trend that should be used in an efficient shock model is that shocks are
quite often dominantly a velocity-dependent device. A simple algebraic shock model is a
polynomial curve fit of the force-velocity characteristic. While this model neglects any
hysteresis that may be present, it adequately represents the general trend of the force-
velocity profile. The current study modifies this algebraic polynomial model slightly,
and uses this algebraic function as an input to a nonlinear low-pass filter to better capture
the lags present in the shock absorber at higher frequency.
2.5 8-Post Testing
Proper suspension tuning is critical for good performance of a race car. The most valid
method to tune suspensions is track testing; however, the amount of available track
testing is extremely limited due to regulations and cost. This has created a need for
alternate methods for suspension tuning. One of the most popular alternate methods is 8-
post testing [2].
During an 8-post test, the tires are supported by 4 actuated wheel platforms, or
wheelloaders, that support the vehicle weight and simulate inputs from the track surface.
Four actuators, or aeroloaders, are attached to the vehicle’s chassis to provide heave
force, roll moment, pitch moment, and warp moment which simulate the effect of inertial
26
and aerodynamic forces present during a track test. The vehicle is shaken on the rig with
drivefiles that simulate track conditions or that are used to characterize the vehicle
response to more generic inputs such as sine waves or broadband signals. Wheelloaders
are typically controlled in position control, while aeroloaders are controlled in force
control. The drivefile is repeated for several suspension setups as sensor measurements
are recorded and the sensor responses are compared between setups. This information is
often used to find or eliminate potential setups prior to track testing or racing. It can also
be used to provide sensitivity information to allow the race engineer to make adjustments
on the track.
Common sensors during a rig test can be broken into rig sensors and vehicle-specific
sensors. The rig sensors available for every test include actuator force, position, and
acceleration which can be used to determine control error, tire force, and chassis position.
Additional instrumentation may be placed on the vehicle as desired. These
measurements typically focus around the suspension, tires, and chassis positions.
Common vehicle instrumentation includes suspension displacement; suspension forces
such as spring forces, shock forces, and bump stop forces; wheel accelerometers; and
chassis positions at critical locations on the car.
Track-based drivefiles are typically generated using an iterative process called drivefile
identification or drivefile iteration, which is described by Kelly in [1]. Drivefile iteration
is a process that applies sensor data from a track test session to generate an 8-post rig
drivefile that represents a particular track. First, the vehicle is tested on the desired track,
while desired sensor measurements are recorded. The vehicle is then installed on the 8-
post rig with the same sensors and a transfer function H is developed between the rig
actuators and the vehicle sensors. The vehicle is then shaken with an initial guess at the
drivefile X, and the error is calculated between the rig sensors Ya and the track sensors Yd.
The initial drivefile is then updated in the frequency domain using
( )1update a dX X KH Y Y−= + − (2.7)
Where K is a step size for the update. The new drivefile is run on the rig, and the
drivefile is updated again. This process is repeated until the error reaches an acceptable
27
level. The identified drivefile is then used to “replay” the track for all future shock
setups.
The vehicle response on the rig is analyzed in different ways. The most basic method is
to simply look at the sensor time traces, power spectra, or frequency responses and
compare them between runs. This will give a detailed picture of the response, and is best
used to investigate a specific detail of the vehicle response. Since there are often many
sensors and runs, it can be difficult to understand the general trends of the test by looking
at time traces.
To get a summary of the trends in the test, the sensor signals are often used to calculate a
series of RMS-based response measures. Each response measure is a scalar statistic that
is calculated from the data vectors. Two common RMS-based response measures are the
“Grip Number”, which is based on the RMS of the total tire force, and the “Aero
Number”, which is based on RMS of the ride height variation. The response measures
are often broken down by lap markers and can be bandpass filtered to highlight a desired
frequency range. Once the response measures are calculated, they can be compared
between runs or as a function of the varied suspension parameters.
While this analysis only considers a small portion of the dynamic behavior of the vehicle
on the 8-post rig and does not address differences between the vehicle dynamics on the
rig and on the track, this method has been used with positive results. Due to these
limitations, the interpretation of results from 8-post rig tests has largely been
experienced-based, making it prone to error when conditions occur outside the test
engineer’s experience base.
While rig testing has become a popular tool in the racing industry in the past few years,
even the best teams struggle with how to successfully use rig data to improve lap times
[2]. Three of the major issues in 8-post testing today include
1. Drivefile Development: How to take track data and use it to quickly generate a
drivefile that can accurately simulate the track,
28
2. Data Analysis: How to relate data from a laboratory testing rig with non-rolling
tires, sitting on teflon pads, with how the vehicle will respond on a race track, and
3. Experimental Design: How to run the experiments to efficiently determine the
best setups and the sensitivity of the response to different setup parameters.
This study focuses on issue 3, which has been discussed in Chapter 1. Another critical
area of research in 8-post rig testing is dealing with inconsistencies between the vehicle
excitation and behavior between the track and laboratory environments which may lead
to false trends observed on the 8-post rig. A comparison of some of these differences is
summarized in Table 2.1.
Table 2.1. Comparison of vehicle excitation and behavior on track and on 8-post rig
Road Inputat time t
On Track On 8-Post
A distribution of heights that interacts with the tire contact patch
A single “equivalent” wheelloader displacement
Driving Line- Road under tires- Acceleration
Driver will seek ideal driving line, which depends on setup
Constant
Aerodynamic Forces
A function of the chassis position, which changes with setup
Determined once during drivefile iteration
Tire DynamicsRolling, stressed, heatedOn asphalt or concrete
Stationary on Teflon pad
Anti-Features / Jacking Forces
Braking, acceleration, and cornering interact with vertical motion due to suspension geometry
Lateral/longitudinal acceleration minimalBrakes and driving torque not applied
Road Inputat time t
On Track On 8-Post
A distribution of heights that interacts with the tire contact patch
A single “equivalent” wheelloader displacement
Driving Line- Road under tires- Acceleration
Driver will seek ideal driving line, which depends on setup
Constant
Aerodynamic Forces
A function of the chassis position, which changes with setup
Determined once during drivefile iteration
Tire DynamicsRolling, stressed, heatedOn asphalt or concrete
Stationary on Teflon pad
Anti-Features / Jacking Forces
Braking, acceleration, and cornering interact with vertical motion due to suspension geometry
Lateral/longitudinal acceleration minimalBrakes and driving torque not applied
Due to the highly competitive nature of racing, very little of the development work done
in 7-post testing has been published. There have been 3 technical papers with data from a
7-post rig test that are applicable to this study, each giving a small piece of information
about how 7-post rigs are used for suspension development. Since there is so little
published in this area, the key details of these three papers are shown below.
Kelly et al. discusses the process they apply to perform drivefile iteration on the 7-post
rig at the Automotive Research Center (ARC) in Indianapolis [1]. This process is
described above. This paper performed drivefile iteration on a Champ car using track
data from Sebring raceway. The wheel hub accelerometers were bandpass filtered from
1-25 Hz to iterate the wheelloader drivefile, and the lateral acceleration, longitudinal
29
acceleration, and vehicle speed were used to determine the aeroloader drivefile. The
drivefile error converged from 40% RMS error to 20% RMS error in 8 iterations.
The paper suggests analysis of suspension setups using both frequency response
functions and drivefile playback. Swept sine testing can be used to make sure the
damping levels are acceptable for the stiffness levels. Track simulation can be used for
further investigation of how the car will respond to specific track events.
Miller reported on a 7-post test session with Clemson University’s Formula SAE car on
Öhlins’ 7-post rig [23]. This test session conducted two studies, first to tune the anti-roll
bars (ARBs), and second to tune the shocks. Initial testing indicated a large amount of
friction in the pushrod, bellcrank, and A-arm bearings. To tune the ARBs, a drivefile was
used which slowly rolled the chassis to the right using the aeroloaders. The front and rear
ARBs were varied in a 3x3 matrix, and the load transfer distribution was used to
determine the best ARB settings. Once the best ARB settings were determined, they
were used for all future tests. Next, another drivefile was used to tune the shock
compression adjustment, by cycling the wheelloaders in heave at 10 in/s peak velocity,
while the aeroloaders were used to simulate 1.4 g cornering acceleration and 0.2 g
longitudinal acceleration. The shock compression adjusters for the front and rear were
also varied in a 3x3 matrix in low, medium, and high settings. The data was used by the
Öhlins engineers to calculate response measures for each test. The post-processing steps
used to calculate these measures from rig data were not described. These measures
included mean left grip disturbance, mean right grip disturbance, mean overall grip
disturbance, roll plot, LF grip disturbance, RF grip disturbance, LR grip disturbance, RR
grip disturbance, body heave, body pitch, and body roll. These response measures were
used to create contour plots for the ARB matrix and the shock matrix, which provides a
summary of the sensitivity to the adjustments.
Kowalczyk uses a linear 7-degree-of-freedom vehicle model to simulate the vehicle
response on the rig for the purpose of helping focus rig data analysis, in particular to
identify tradeoffs and determine realistic limitations of tuning [24]. A chirp drivefile was
used for the wheelloaders in heave and pitch, with the amplitude envelope adjusted to
30
match track data power spectral density (PSD). Contour plots were generated for the
peak of the power spectrum for heave and pitch acceleration for different front and rear
shock settings. A tradeoff between optimizing heave acceleration and pitch acceleration
was shown, which forces the race engineer to decide if the potential benefit in improving
one metric outweighs the other. They chose to minimize pitch variations, which would
reduce the variations in the distribution of normal force variation front to rear. The
power spectrum from rig data was then presented which showed a significant
improvement in pitch response, with only a slight degradation in heave response.
Seven-post testing for motorsports applications originated from 4-post testing which was
primarily used for comfort and noise vibration harshness (NVH) applications. One of the
earliest developments for motorsports testing was to put the car on a 4-post rig and attach
soft springs to the chassis. These springs were preloaded to simulate a DC aerodynamic
downforce. In 1994, Kasprzak performed testing on one of these modified 4-post rigs
and compared it to simulation data [7]. The simulations first used a 4th-order linear half-
car model, and later substituted the linear shocks with a lookup table using shock
dynamometer data from a sine test. Kasprzak states that the model trends show good
correlation with experimental data. Once promising setups were found using the
simulations, they could be tested on the rig and the setup could be fine-tuned. They used
this process for a race team, and the team reported the setup was better.
The last two papers by Kowalczyk and Kasprzak have suggested that linear models have
been successfully applied to predict trends and accelerate setup tuning on 7-post rig tests.
Linear models are also used in an iterative process to develop drivefiles. These trends
can be used to select potential setups to be tested on the rig, identify parameter sensitivity
for tuning, and show tradeoffs and realistic limitations to suspension tuning.
Unfortunately, both of these papers presented limited rig data and did not quantify the
accuracy of their linear models.
Another thing that was not described in these papers is how they developed their vehicle
models. Having an accurate vehicle model and knowing the limitations of the models
can be critical to successfully applying simulation to help rig testing.
31
Chan introduced a 12 degree-of-freedom nonlinear vehicle model for virtual 7-post
testing, which includes tire separation and large rotations, for use in future 7-post
research, including identification studies [25]. While this model accounts for things that
are not possible with a linear model, it does not account for linear chassis compliance that
is present in most oval and road racing cars, and it does not account for the coupling
present in a solid-axle suspension.
2.6 The Role of Experimental Data in Vehicle Modeling
To create an accurate model with known performance limitations, it helpful to use
experimental data to validate the model accuracy. If desired, the model can also be
updated to better match the experimental data. This process of model validation and
updating can be performed on the component level, the system level, or both depending
on the application. Some relevant examples showing trends in applying experimental
data to model validation and updating are shown below.
In the automotive industry, vehicle models are quite commonly used early in the
development cycle when vehicle test data is not available. This makes model updating
using vehicle system data difficult, so the focus is often on integrating validated
component models. Vilela correlated proving ground data collected for a selection of
different proving ground tests with a detailed ADAMS model assembled using
component models [26]. The correlation was determined to be acceptable, indicating that
the model could have been used before the vehicle was manufactured.
Often, system level data for an existing vehicle is used to refine an existing vehicle
model, which will be used in future studies. The experimental data is often assumed to
be the ideal response for the model to try to achieve, although experimental errors can
make this a poor assumption. One basic method to use test data to update models is to
first perform a parametric study to determine the sensitivity of error to model parameters.
The model parameters can then be adjusted using the sensitivity information to reduce
modeling error [27].
32
A more systematic method for model updating is to apply optimization methods to find
parameter values that minimize the model error. Hu performed model validation and
updating on a half-car ride model using proving ground data [28]. The initial guess at the
parameters was defined using component tests, which were then updated to better match
the proving ground data using optimization of modeling error.
While many studies use proving ground data for model development, there has been an
increasing trend to use laboratory test data since the experiments are run in a more
controlled environment with higher repeatability [29]. For example, Cheli and Sabbioni
performed 4-post testing on a van to identify the unknown stiffness, damping, and inertial
parameters [30]. The parameters were optimized to best fit the experimental data using
nonlinear optimization. Further data was used to show that there was acceptable
correlation between the model and data. The ability of the model to predict the effect of
changing parameters was not discussed. Andersen performed parameter identification for
a nonlinear multibody dynamics model of a McPherson strut suspension using laboratory
data from a quarter-car rig [31].
Most model updating studies attempt to match the displacements and/or accelerations of
the sprung and unsprung masses. One additional sensor measurement that is important
for an 8-post model to predict is the tire normal force. Ziegenmeyer optimized the
parameters of a linear quarter-car model to minimize the prediction error for a quarter-car
rig test on a quarter-car rig which included real suspension components and a tire sitting
on a wheel platen [32]. The model was then used to predict the accelerations of the
sprung and unsprung masses and the tire normal force with reasonable accuracy.
The above studies use optimization methods to update model parameters that are highly
dependent on gradient estimates of the error as a function of model parameters.
Depending on the model structure, this error function can have local minima, and the
optimization method may converge to these local minima. This can be further
complicated with experimental data with noise and nonlinearities. The success of these
optimization methods can also be dependent on a good initial guess, which requires
accurate component models. Also, gradient methods are not applicable when the
33
parameters can only be chosen from a discrete space. Due to these issues, alternative to
gradient-based optimization is often desired.
For example, Alasty generated target data using an ADAMS simulation and applied
genetic algorithms to find a global minimum to the error function of a nonlinear seven
degree of freedom model [33]. He also discusses identifiability conditions for nonlinear
models about an operating point.
The above studies all use structural models (both linear and nonlinear), where the models
have been constructed using components that have a specific physical meaning. Many
applications do not require accurate estimation of the structural parameters, only a model
of the input-output behavior. If this is the case, a more generic linear or nonlinear
dynamic model structure can be used, which has more modeling freedom and can be
identified using generic system identification tools. By using a generic input-output
model structure, there is no need for prior knowledge of the geometry of how
components in the physical system are arranged or the boundary conditions for their
connections.
For example, if the system operates within a range in which the dynamics are linear, the
system may be represented by many different linear dynamic models. General linear
dynamic models can be characterized using frequency responses or impulse response,
which can be identified easily using least-squares regression techniques. If a transfer
function or state-space model is required, a variety of system identification methods can
be applied to fit these types of models [34].
34
Chapter 3 Shock Absorber Modeling
This chapter develops a physically-motivated nonlinear dynamic shock absorber model
that can be quickly fit to experimental data and implemented in simulation studies. This
model is based on the understanding that the shock is dominantly a velocity-dependent
device, with lag due to compressibility effects.
The model consists of an algebraic backbone, which is a function of velocity alone, and a
nonlinear low-pass filter, which has been designed based on the observation that shocks
often exhibit less hysteresis at higher velocities. Due to the simplicity of the model, it
can be fitted with data and evaluated quickly.
The next section presents a simple physical shock absorber representation that motivates
several shock models. This is followed by a description of the experimental procedure
for shock dynamometer testing. These models are then fitted using the experimental data
for 12 different shock configurations. After the models have been fitted to the
experimental data, they are then validated using both broadband and sine wave inputs.
This analysis allows us to specify the conditions under which the model can accurately
predict force and where the model may need some improvement.
3.1 Physical Motivation
This section describes the physical motivation for the shock models presented later in this
chapter. First, a simple linear model for the internal flows inside a shock absorber is
presented. Using this model, both the incompressible and compressible equations are
derived. The fluid dynamic equations derived for the incompressible and compressible
cases are then represented as mechanical system equivalents, which is a more intuitive
form for a component model in a mechanical system.
35
The equivalent mechanical system is found to be a damper in series with one spring,
representing compressibility effects, and in parallel with another spring, which represents
the resistance from the gas charge to the rod entering the rebound chamber.
3.1.1 Physical Model of the Shock Absorber
A simplified construction of a shock absorber is shown in Figure 3.1. The construction
includes a shock body, a piston-rod assembly, and a floating piston. The piston separates
the compression and rebound chambers which are both filled with hydraulic fluid, while
the floating piston separates the fluid in the compression chamber from the nitrogen
charge in the nitrogen chamber. When compressed, the piston-rod assembly moves up
into the shock body, which changes the volume of the compression and rebound
chamber. Since the rod must be inserted into the shock body, the fluid displaced by the
rod insertion must be compensated for by motion of the floating piston. The change in
volume in the rebound chamber causes flow through the bleed holes and shims stacks in
the main piston. This flow creates a resistance to motion that is a function of velocity.
Nitrogen Reservoir
Floating Piston
Piston
Rebound ChamberOil
Nitrogen
Oil
Shock in Compression
Compression Chamber
Oil
Nitrogen
Oil
Shock in Compression
Nitrogen Reservoir
Floating Piston
Piston
Rebound ChamberOil
Nitrogen
Oil
Shock in Compression
Compression Chamber
Oil
Nitrogen
Oil
Shock in Compression
Figure 3.1. Basic shock absorber construction
Another method of visualizing the fluid circuit inside the shock absorber is the analogous
fluid circuit shown in Figure 3.2. This explicitly shows each effect as an absolute fluid
flow, which allows the engineer to visualize the shock flows and gain insight on the
36
equations governing shock function. If inertial effects are neglected, only relative flow
rates matter. To simplify the fluid system, first we fix the main piston in the shock body.
We then represent the effect of moving the piston in the shock body by changing the
volumes of the compression and rebound chambers the same amount that they are
changed by moving the piston-rod assembly in the shock. The effect of the nitrogen
reservoir can be modeled as a spring-loaded accumulator.
Ac
Ar
Velocity
Qp
Nitrogen Reservoir
Compression Chamber
Compressibility
ReboundChamber
Compressibility
Compression Chamber Volume Change
Rebound Chamber Volume Change
pc
pr
kg
kc
kr
Ar
Ac
Qg
QcQc
Qr
Qr
Ac
Ar
Velocity
Qp
Nitrogen Reservoir
Compression Chamber
Compressibility
ReboundChamber
Compressibility
Compression Chamber Volume Change
Rebound Chamber Volume Change
pc
pr
kg
kc
kr
Ar
Ac
Qg
QcQc
Qr
Qr
Figure 3.2. Equivalent fluid system for shock absorber
When the fluid is incompressible, conservation of mass reduces to conservation of
volumetric flow rate. Conservation of volumetric flow rate is easy to visualize in a fluid
circuit, as the net flow for any control volume must be zero. When the fluid is
compressible, conservation of volumetric flow rate no longer applies. To approximate
the incompressible system using a simplified incompressible system, the error between
conservation of volume and conservation of mass can be approximated using an
equivalent flow to an accumulator. This model is consistent with a compressible fluid
with constant bulk modulus. The equivalent flow due to compression can be thought of
as fluid volume and energy storage due to fluid compression. This allows us to treat a
more complex compressible system like a simple incompressible system and view fluid
compression as a flow compressing a spring.
37
The rate of change of volume for the compression and rebound chambers is
c cQ A v= (3.1)
r rQ A v= (3.2)
where v is the rod velocity, Qc and Qr are the rate of volume change in the compression
and rebound chambers, and Ac and Ar are the areas on the main piston on the compression
and rebound side. In Figure 3.2, the changes in volumes are caused by controlling two
pistons, which displace the correct amount of fluid. When the piston on the compression
side causes a flow to the compression chamber, the flow must displace the floating
piston, pass through the main piston, or go to the compression chamber compressibility.
Similarly, as the rebound chamber expands, flow must come from the main piston or
from fluid expansion. This gives continuity equations for the compression and rebound
chambers of
c g p cQ Q Q Q= + + (3.3)
p r rQ Q Q= + (3.4)
Where cQ and rQ are the effective compressive flows for the compression and rebound
chambers, while pQ is the flow through the main piston and gQ is the flow that displaces
the floating piston.
For the purpose of analysis, we will model the accumulators as linear springs and the
piston flow as a linear flow resistance. The gas charge will behave like a linear spring for
small deflections or low gas pressure, while the fluid compressibility will act like a linear
spring if the bulk modulus is constant. The piston flow resistance for a constant area
orifice can be modeled as linear for laminar flow or small velocities. Linear flow
characteristics can also be achieved in other conditions by the opening of the shim stacks,
which effectively increases the orifice area with velocity. We will extend the results to
motivate the development of nonlinear shock models in a later section. The constitutive
equations for the accumulators and main piston are
38
Compression Chamber: 1cc ckQ p= (3.5)
Rebound Chamber: 1rr rkQ p= (3.6)
Gas Chamber: 1gg ckQ p= (3.7)
Main Piston: c r pp p CQ− = (3.8)
where cp and rp are the pressures in the compression and rebound chambers, ck and rk
are the stiffnesses for the compression and rebound chamber compressibility, gk is the
stiffness of the gas charge, and C is the flow resistance for the main piston.
The shock force can be found by drawing a free-body diagram of the piston-rod
assembly. This yields the shock compression force as
c c r rF p A p A= − (3.9)
This model neglects losses due to piston friction and also kinetic energy of the fluid and
shock components. These equations are further developed for the incompressible and
compressible cases in the following sections.
3.1.2 Incompressible Model
When the fluid is incompressible, the effective compression and rebound flows due to
compression are zero. Combining Equations (3.1) – (3.8) yields
( )1 1gc c c rk CA v p p p= + − (3.10)
( )1c r rC p p A v− = (3.11)
Substituting (3.11) into (3.10) and integrating allows us to solve for the compression
pressure
c g rod cop k A x p= + (3.12)
39
where cop is the initial fluid pressure, x is the piston rod displacement, and rodA is the
area of the rod. Substituting the compression pressure into Equation (3.11) yields the
rebound chamber pressure
r c rp p CA v= − (3.13)
Now that we have the two chamber pressures, we can use them to find the shock force
( ) ( )2 2r g rod co rod
inc inc o
F CA v k A x p A
C v K x F
= + +
= + + (3.14)
where incC , incK , oF are constants that define the shock model’s damping, stiffness, and
nominal force properties at the system level. This shows that the incompressible model
force can be represented as a linear damper in parallel with a linear spring and an offset
force.
3.1.3 Compressible Model
Combining Equations (3.1) – (3.8) yields
( ) ( )
( ) ( )
1 1'
1 1
g
r
c c r ck C
r c r rk C
p p p A v
p p p A v
= − − +
= − − (3.15)
Where 'gk is stiffness of the gas spring and the compression chamber spring
mechanically in series. While 'gk will usually be finite due to the gas charge, rk may be
infinite if the rebound chamber is incompressible. Assuming that both 'gk and rk are
finite, we may solve for the pressure change rates and add Equation (3.9) to form the
state-space model
40
[ ]
' ' 'g g
r r
k kc c g cC C
k kr r r rC C
cc r
r
p p k Av
p p k A
pF A A
p
−= + −−
= −
(3.16)
Rewriting the model as a transfer function yields
( )( ) ( ) ( )
( )'' 2 2 22
' ''
'
'
1
g rg r
g r g rg r
g r
k kk k Cc g r r rodk k k krod k k
Ck k
A k A k AAF sV s s
+ ++
+
+ + = ++
(3.17)
This equation can be further simplified by assuming the gas stiffness is much softer than
the rebound and compression stiffnesses and that the compression, rebound, and rod areas
are similar in magnitude. This assumption simplifies Equation (3.17) to
( ) ( ) ( )2 211
11
r
r g rod oCk
inc inc o
F CA v k A x Fs
C v K x Fsτ
= + + + = + + +
(3.18)
Where incC , incK , and oF are the damping rate, spring rate, and initial force found for the
incompressible case. This shows that the only difference between the incompressible and
the compressible models is a first-order low-pass filter on the shock force. The lag, τ , is
caused by the compressibility in the rebound chamber. This provides the motivation for
the shock models that are developed in Section 3.15.
3.1.4 Equivalence to Mechanical Systems
Shocks are often modeled as an ideal dashpot in series and/or in parallel with springs.
Modeling a shock absorber using mechanical elements is a natural fit when the shock is a
component in a mechanical system. In this section we show that the incompressible
model is equivalent to a damper in parallel with a spring and in the compressible case, the
model is equivalent to a damper in series and in parallel with springs.
41
c kp
x ckp
ks
x
y
(b)(a)
FF
c kp
xc kp
x ckp
ks
x
y
ckp
ks
x
y
(b)(a)
FF
Figure 3.3. Mechanical equivalents: (a) Damper in series with a spring,
(b) Damper in series and in parallel with springs
The equation for a damper in parallel with a spring as shown in Figure 3.3a is
pF cx k x= + (3.19)
This is in direct agreement with the incompressible model of Equation (3.14), with
2
2r
p g rod
c CA
k k A
=
= (3.20)
The equation for a damper in series and in parallel with springs as shown in Figure 3.3b
is
( ) 1s
pck
cF x k xs
= + +
(3.21)
This is in direct agreement with the compressible model of Equation (3.18), where
2
2
2
p g rod
r
s r r
k k A
c CA
k k A
=
=
=
(3.22)
The previous sections have derived a linear compressible shock absorber model. This
work will next be applied to develop models for real shocks.
42
3.1.5 Extension of Physical Motivation to Shock Models
While the above models are linear, shock absorbers are inherently nonlinear devices.
This section extends the above linear discussion to modeling real shocks. A common
nonlinear shock model is a simple curve fit of the force as a function of velocity
( )sF f v= (3.23)
The curve fit is often a polynomial, but some studies have used transcendental shape
functions. An example force-velocity plot and a curve fit model are shown in Figure 3.4.
While algebraic curve fits do not capture frequency-dependency, they capture the general
shape of the force-velocity response and are an excellent starting point for shock
modeling.
-40 -30 -20 -10 0 10 20 30 40-1000
-500
0
500
velocity, in/s
Forc
e, lb
s
DataCurve Fit
-40 -30 -20 -10 0 10 20 30 40-1000
-500
0
500
velocity, in/s
Forc
e, lb
s
DataCurve Fit
Figure 3.4. Sample shock dynamometer data and curve fit
The first approach that was tried for an algebraic model was a polynomial curve fit.
Since the force-velocity plot often has a complex shape, a high-order polynomial was
used before increasing the polynomial order produced no noticeable reduction in
modeling error. Because a high-order polynomial was required to capture the complex
shape, significant errors occur if the model is used to predict a force outside of the fit
data. For one model, a velocity less than 5% outside of the range of the fit data caused
more than 10,000 lb of error, as shown in Figure 3.5a.
43
While a high-order polynomial is useful to drive down the error, it cannot be used for
extrapolation. Further examination of multiple shock curves yielded the observation that
the shape of the shock curve at high velocity is often close to linear and the high-order
polynomial is only needed at low velocity. This observation prompted the modification
to fit a high order polynomial at low velocities and fit a low order polynomial at high
velocity compression and another low-order polynomial at high velocity rebound, as
shown in Figure 3.5b. In the transition regions between low and high velocities, the two
polynomials are linearly combined to provide a continuous transition. We label this
algebraic shock model as “Polylinear”.
-60 -40 -20 0 20 40 60 80-1500
-1000
-500
0
500
1000
1500
velocity, in/s
Forc
e, lb
s
Low-OrderFit
Low-OrerFit
High Order
Fit
v1 v2
-60 -40 -20 0 20 40 60-1500
-1000
-500
0
500
1000
1500
velocity, in/s
Forc
e, lb
s
modeldata
(a) (b)
-60 -40 -20 0 20 40 60 80-1500
-1000
-500
0
500
1000
1500
velocity, in/s
Forc
e, lb
s
Low-OrderFit
Low-OrerFit
High Order
Fit
v1 v2
-60 -40 -20 0 20 40 60-1500
-1000
-500
0
500
1000
1500
velocity, in/s
Forc
e, lb
s
modeldata
(a) (b)
-60 -40 -20 0 20 40 60 80-1500
-1000
-500
0
500
1000
1500
velocity, in/s
Forc
e, lb
s
Low-OrderFit
Low-OrerFit
High Order
Fit
v1 v2
-60 -40 -20 0 20 40 60-1500
-1000
-500
0
500
1000
1500
velocity, in/s
Forc
e, lb
s
modeldata
(a) (b)
-60 -40 -20 0 20 40 60 80-1500
-1000
-500
0
500
1000
1500
velocity, in/s
Forc
e, lb
s
Low-OrderFit
Low-OrerFit
High Order
Fit
v1 v2
-60 -40 -20 0 20 40 60-1500
-1000
-500
0
500
1000
1500
velocity, in/s
Forc
e, lb
s
modeldata
(a) (b)
Figure 3.5. (a) Error in extrapolating a high-order polynomial fit,
(b) Regions for polynomials in Polylinear model
While the algebraic curve fit model typically captures the general trend of the response, it
cannot capture the dynamic behavior. This is particularly important in shocks where the
hysteresis is significant. For example, the shock data in Figure 3.4 shows almost 300 lb
hysteresis, which is 20% of the 1500 lb total force range. If we want an accurate
simulation, we need to capture the shock’s dynamics.
In the previous sections, we have established that the behavior of the linear compressible
shock absorber model shown in Figure 3.2 was equivalent to a linear dashpot with
springs both in series and in parallel. Since race shocks often run low gas spring rates,
we will drop the gas stiffness term for now. The model in differential equation form is
44
( )1sF F Fτ= − + (3.24)
where sF is the algebraic force the shock would achieve if there was no compliance in
series or at low frequency. The term s
ckτ = acts as a system time constant. We can
replace the proportional shock force sF cx= with the nonlinear algebraic shock force
calculated by the curve fit model in Equation (3.23). The resulting dynamic shock model
takes on the structure shown in Figure 3.6. This model includes a linear first-order
differential equation, and a nonlinearity on the velocity input. We label this shock model
as “LPF1”.
( )vfFs =v
velocity
Polylinear Model
FsPolylinear
Force
FLPF1Force
LPF1 Shock Model
First-Order Filter
( )1sF F F
τ= − +( )vfFs =
vvelocity
Polylinear Model
FsPolylinear
Force
FLPF1Force
LPF1 Shock Model
First-Order Filter
( )1sF F F
τ= − +
Figure 3.6. LPF1 shock absorber model
The simple first-order lag model may work reasonably well for some shocks, but many
shocks only exhibit hysteresis at lower velocities when the flow is changing directions,
pressures in the compression and rebound chambers are just starting to develop, and the
valves are starting to open. At higher velocities, the valves are fully open and the fluid
has been compressed, so the force-velocity plots often exhibit very little hysteresis and
can be accurately modeled using an algebraic model. When using the LPF1 model, the
lag causes hysteresis at low velocity where it is needed, but it also causes lag at higher
velocities where zero lag is best.
This phenomenon may be explained by extending our physical shock model to include
fluid stiffness that changes as a function of velocity. We then find that the time constant
for the linear model is s r
c Ck kτ = = . High lag at low velocities and no lag at high velocity
suggest that the fluid stiffens at higher velocity and behaves more like an incompressible
fluid, as shown in Figure 3.7. At low velocities, the fluid has not changed volume much
and the fluid has a nominal compressibility, giving the shock a nominal time constant. At
45
higher velocities, the fluid has already been compressed and is almost incompressible,
making the time constant almost zero. This suggests that the fluid compressibility is
more accurately represented as a stiffening spring, not a linear spring.
velocity
τ =
k r/C
volume change
Forc
e
(a) (b)
Compressible Range Compressible Range
velocity
τ =
k r/C
volume change
Forc
e
(a) (b)
Compressible Range Compressible Range
Figure 3.7. The effect of a stiffening fluid on time constant:
(a) Fluid stiffness, (b) Time constant
It would seem that all that would be needed to model this behavior is to make the time
constant in Equation (3.24) a function of velocity, with a curve similar to the one shown
in Figure 3.7b. While this is possible if there is significant lag at high velocity, many
shocks show very little hysteresis at high velocity. A small time constant at high
velocities would require small time steps for accurate numerical integration. This will
cause slow simulations, so it is not an acceptable solution.
A more efficient way to get high lag at low velocity and no lag at high velocity is to use
the lagged force at low velocity and use direct feedthrough of the algebraic force at high
velocity. The model equations are
( )( ) ( )( )
1
1
lag s lag
s lag s s
F F F
F F F F Fτ
α α
= −
= + − (3.25)
where the weighting function α is unity at low velocities, zero at high velocities, and a
linear combination of the two between low and high velocities, as shown in Figure 3.8.
46
This model consists of a linear first-order differential equation and nonlinearities at both
the input and output. We label this model “LPNL2”.
( )sFα
sF
1
1F 2F
UseLagged Force
UseCurve
Fit
( )sFα
sF
1
1F 2F
( )sFα
sF
1
1F 2F
UseLagged Force
UseCurve
Fit
Figure 3.8. Output weighting for LPNL2 model
3.1.6 Summary
The purpose of this section was to motivate the development of physically-based shock
absorber models. We started by developing a linear compressible shock absorber model.
The equations were solved for both the incompressible and compressible cases. Next,
mechanical equivalents were found for both the incompressible and compressible models.
In the case of the incompressible model, the mechanical equivalent is a damper in parallel
with a spring that represents the gas charge. In the case of the compressible model, the
mechanical equivalent is a damper in series with a spring that represents the rebound
chamber compressibility and in parallel with a spring that represents the gas charge.
The physical insight gained by developing the linear models was then extended to define
physically-motivated shock absorber models. While the incompressible linear shock
model was a linear function of velocity, real shocks are nonlinear. To capture the shock
force as a nonlinear function of velocity, we defined a Polylinear model, which curve fits
a high-order polynomial at low velocities and a low-order polynomial at high velocities.
This was motivated by the fact that the while the shock curve may have a complex shape
at lower velocity, the shock often exhibits near-linear behavior at higher velocities.
Having a low-order curve fit at high velocities also has the advantage of being well-suited
for extrapolation to velocities beyond the original curve fit data without significant error.
47
While a curve fit of the force-velocity plot captures much of the shock response, shocks
often exhibit significant hysteresis which must be captured in any accurate shock model.
Motivated by our linear compressible shock model, our first dynamic shock model
(LPF1) was just the curve fit filtered by a first-order lowpass filter.
The observation was then made that shocks typically only exhibit hysteresis at low
velocities, while at higher velocities there is almost no hysteresis to the force-velocity
curve. Unfortunately, the LPF1 model adds thickness at all velocities. This motivated a
dynamic shock absorber model (LPNL2) which used the lagged force of LPF1 at low
velocities and the Polylinear curve fit at high velocities.
While this model was developed for a shock absorber commonly used in motorsports
applications, the minimal assumptions that led us to arrive at this model structure allow it
to be applied to a wider class of dynamic systems. First, we assume the force output is
dominantly an algebraic function of the velocity input. Next, the high velocity behavior
can be described by a low-order polynomial. The dynamic behavior can be approximated
by a nonlinear stiffening spring in series with a damper, producing lag and attenuation at
low velocity, but not at high velocity, where the spring is effectively rigid. In our case,
the stiffening effect is a result of a fluid that has a stiffening bulk modulus, but it could be
used to describe other stiffenening compliances in series with a damper, such as
compliant rubber bushings. It may also be possible to accommodate other dynamic
behavior by changing the weighting function.
To fit the three model structures developed in this section to the desired shocks, the
shocks are first tested on a shock dynamometer and then the data is used to fit the models
and to validate that the model accurately represents the shock performance under
different conditions than the fit data to avoid overfitting. These topics are covered in the
following sections.
48
3.2 Experimental Setup
This section describes the shock testing conducted to collect the data necessary to fit the
shock models described in the previous section and to validate the model’s ability to
predict forces. We start by describing the Roehrig EMA shock dynamometer used for
testing. We continue by describing the 12 different shock configurations we tested. We
conclude by describing the drivefiles that were used for model fitting and validation of
the shock models.
3.2.1 Roehrig EMA Dynamometer
To develop an empirical shock absorber model, a Penske 7300 NASCAR Nextel Cup
shock absorber was tested on a Roehrig Electromagnetic Actuator (EMA) shock
dynamometer, as shown in Figure 3.9. This dynamometer uses an electromagnetic
actuator to actuate the shock. This allows a wide variety of inputs, including sine waves,
triangle waves, random inputs, track data, or any user-defined input.
Figure 3.9. Penske 7300 shock absorber on a Roehrig EMA (photo by author, 2009)
The Roehrig EMA consists of a load cell located at the top of the dynamometer, an
electromagnetic actuator located at the base of the machine, and an infrared thermocouple
49
to measure shock body temperature. The system is driven by two electromagnetic
actuators that act on a ram connected to the lower shock mount. The Roehrig EMA is
capable of forces up to 2000 lb, 8 inches of stroke, excellent frequency response up to 80
Hertz, and speeds of 100 in/s [35].
Tests on the EMA are controlled using Roehrig’s Shock software. This software allows
test definition from both standard waveforms and user-defined inputs. Test signals are
defined about a user-defined DC offset, which was set to 3 inches in this study to put the
shock absorber near the middle of its range of travel. In addition, Shock allows a warm-
up sequence to be run prior to each test session, where the shock is run until it reaches a
specified temperature. This allows for more uniform testing conditions. For this study,
the shock was warmed up to 110oF prior to each test. During each test, measurements of
shock position, velocity, force, and temperature are recorded at 2,000 Hz. At the end of
each experiment, this data is saved for further analysis.
3.2.2 Shock Configurations
As discussed in Chapter 2, each Penske 7300 shock is designed to be highly adjustable
and can be rebuilt with different internal components, giving the shock engineer
flexibility to rebuild the shock and tune the shock characteristics to a desirable setting.
The main internal adjustments are the compression shims, rebound shims, and bleed jet
on the main piston and the head valve. The external adjustments are the bleed adjuster
and gas pressure.
To investigate the effectiveness of the shock models for a range of shock settings, we
made 3 different shock builds. The first build was used as a baseline. The second build
has a softer compression shim stack on the head valve. The third build has a softer
compression shim stack on the main piston. Each build was tested at 50 psi and 150 psi
gas pressure and at a low setting (1 click) and a medium setting (14 clicks) for the bleed
adjuster. A force-velocity plot for the three builds at 50 psi and low bleed setting are
shown in Figure 3.10. A summary of the 12 different configurations is shown in Table
3.1.
50
(a) (c)(b)
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
(a) (c)(b)
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
(a) (c)(b)
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
(a) (c)(b)
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
Figure 3.10. Force-velocity for the three shock builds tested:
(a) Build 1, (b) Build 2, (c) Build 4
Table 3.1. Shock configurations tested
Build Name Description Bleed Adjuster Gas PressureBuild 1 Baseline 1 click 50 psiBuild 1 Baseline 14 clicks 50 psiBuild 1 Baseline 1 click 150 psiBuild 1 Baseline 14 clicks 150 psiBuild 2 Softer HV 1 click 50 psiBuild 2 Softer HV 14 clicks 50 psiBuild 2 Softer HV 1 click 150 psiBuild 2 Softer HV 14 clicks 150 psiBuild 4 Softer HV 1 click 50 psiBuild 4 Softer HV 14 clicks 50 psiBuild 4 Softer HV 1 click 150 psiBuild 4 Softer HV 14 clicks 150 psi
3.2.3 Drivefiles
Each of the shock configurations described above was tested using a variety of drivefiles
on the shock dynamometer. For this study, we used random signals for model fitting and
validation purposes. A random drivefile was generated using a multisine waveform by
first defining the shape of the relative amplitude spectrum of the position command at 10
different points between 0.1 and 20 Hz, as shown in Table 3.2. This shape was chosen to
excite the frequencies of interest while not introducing unrealistic velocities into the
shock absorber.
51
Table 3.2. Definition of relative amplitude spectrum
Frequency AmplitudeHz Relative0.1 11 12 0.83 0.74 0.55 0.38 0.210 0.120 0.0530 0.01
To create a more evenly distributed amplitude spectrum, the spectrum defined in Table
3.2 was interpolated at every 0.1 Hz to define the amplitude spectrum at 300 uniformly
spaced points. The unscaled time signal was then created by summing the amplitude
spectrum
( ) ( )sin 2unscaled i i ii
x t A f tπ φ= +∑ (3.26)
where Ai is the value of the unscaled amplitude spectrum at frequency fi. The phase φi at
each frequency is assigned by MATLAB’s uniformly distributed random number
generator, rand, over the range 0 to 360 degrees. This was used to create a 30 second
random signal. Multiple realizations of the random signal were calculated and the one
with the lowest crest factor was chosen to avoid extreme peaks in the drivefile. The
signal is then rescaled to have the desired peak amplitude on the shock dynamometer.
For our drivefile, the peak amplitude was selected to be 1 inch. This drivefile is then
saved for import to the Roehrig shock dynamometer. A 5-second segment of the
drivefile is shown in Figure 3.11.
52
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
time, seconds
Pos
ition
, in
Figure 3.11. Segment of the random drivefile
In addition to the random drivefile described above, we also tested each of the 12 shock
settings with sine waves over a wide range of amplitudes and frequencies, as shown in
Table 3.3. The purpose of the sine wave tests in this study is to serve as a validation data
set for the model. This method will allow us to see at what amplitudes and frequencies
the model performs well, and where the model could use some improvement [36]. This
information can be quite useful when deciding if a shock model is suitable for a particular
simulation application.
In addition to the random and sine tests that were performed for each shock
configuration, a static test was performed for each combination of build and gas pressure.
The static profile is shown in Figure 3.12. It starts at an initial test offset of 2 inches and
stays there for 20 seconds. It then moves up one inch at 0.2 in/s and then maintains the
new position for 20 seconds. This process is repeated until the shock is moved up 3
inches, at which point the process is repeated in the downward direction until the shock
reaches the initial test offset of 2 inches. The static test gives a spring rating of the shock
at three positions and in two directions, which gives us both the force-deflection curve
and also the amount of hysteresis in that curve caused by directional effects such as seal
drag.
53
Each of the three builds were also tested using a 0.5 inch bump test, with the drivefile
shown in Figure 3.13. The transition speed of the bump was 30 in/s.
Table 3.3. Summary of sine wave tests
Amplitude Velocity Frequency(in) (in/s) (Hz)1 1 0.161 3 0.481 6 0.951 10 1.591 20 3.181 30 4.77
0.5 5 1.590.5 10 3.180.5 20 6.370.25 10 6.370.25 20 12.730.1 2.5 3.980.1 5 7.960.1 10 15.92
0 20 40 60 80 100 120 140 160
0
1
2
3
time, seconds
Pos
ition
, in
Figure 3.12. Static test drivefile
0 50 100 150 200
0
0.1
0.2
0.3
0.4
0.5
time, seconds
Pos
ition
, in
Figure 3.13. Bump test drivefile
54
3.2.4 Summary
This section has presented the experimental procedures used to collect the data that will
be used to fit the shock models described in Section 3.1 and to validate their ability to
predict force under different operating conditions.
Twelve different shock configurations are tested to investigate the effect of head valve
shims, compression shims, bleed adjuster, and gas pressure on the model accuracy. Each
configuration is tested with a random drivefile for fitting and validation. Each shock
configuration is also tested using multiple sine waves for validation. Each combination
of build and gas pressure is tested using a static test to determine the shock’s dependence
on position and how that changes with the direction that the position was approached.
Each build was tested using a bump input with transition velocity of 30 in/s to test the
model accuracy during a bump event.
3.3 Shock Absorber Modeling
The previous section described three different shock models and our experimental
procedure for collecting data to fit and validate the models. This section describes our
efforts to fit the models, while the next section covers the model validation. While each
of the 12 shock configurations is fit to all three models, we only present the details for
fitting shock build 1 with 50 psi gas and 1 click rebound in this section for brevity. We
will summarize the key results for the other fits where informative.
The data used for fitting the models was from seconds 1-10 from the random test. All
shock dynamometer data is filtered using an 8th order Butterworth lowpass filter with
cutoff of 50 Hz using MATLAB’s filtfilt command, which provides zero phase shift and
twice the attenuation by filtering the data forwards and then backwards.
3.3.1 Algebraic Modeling
The Polylinear model presented in Section 3.1.5 was fit using seconds 1-10 of the 30-
second random test. The resulting force-velocity plot is shown in Figure 3.14 and a time
55
plot of the forces and error is shown in Figure 3.15. The model produced good fits with a
15th order polynomial at low velocity and a 3rd order polynomial at higher velocities. To
ensure that this polynomial model was not overfitting the fit data, it was also verified that
the error in the validation data was also being reduced as model order increased. The
transition from low to high velocity was set at 15 in/s and the width of the transition was
set to 0.5 in/s. The polynomial model is fast – the 9-second segment was fit in 0.12
seconds and evaluated in 0.02 seconds. The Polylinear model fit was shown previously
in Figure 3.4. Due to significant hysteresis in the force-velocity plot, there was
significant error in the Polylinear model by as much as 29 lb RMS error. This
corresponds to 8% of the signal standard deviation of 366.5 lb.
-40 -30 -20 -10 0 10 20 30 40-1000
-500
0
500
velocity, in/s
Forc
e, lb
s
DataCurve Fit
-40 -30 -20 -10 0 10 20 30 40-1000
-500
0
500
velocity, in/s
Forc
e, lb
s
DataCurve Fit
Figure 3.14. Polylinear model, force-velocity plot
56
0 1 2 3 4 5 6 7 8 9-1000
-500
0
500
time, seconds
Forc
e, lb
s
ModelDataError
0 1 2 3 4 5 6 7 8 9-1000
-500
0
500
time, seconds
Forc
e, lb
s
ModelDataError
Figure 3.15. Polylinear model, error vs. time
3.3.2 LPF1 Model
The first-order LPF1 filter was fit by trying different time constants ranging from 0.1 to
10 milliseconds in 0.1 millisecond increments and selecting the one with the least error.
The resulting force-velocity plot produced by the LPF1 model is shown in Figure 3.16
and a time plot of the forces and modeling error is shown in Figure 3.17. The best time
constant was found to be 1.3 milliseconds, and the resulting LPF1 model produced 8.1 lb
of RMS error. This is a 72% reduction in error from the Polylinear curve fit, which
produced 29 lb of error. This corresponds to 2.2% of the signal standard deviation of
366.5 lb. This clearly indicates that the LPF1 provides a significant improvement in
modeling error over a simple force-velocity curve fit. Further inspection of the validation
data for the LPF1 model will confirm this result in Section 3.4.
57
-40 -30 -20 -10 0 10 20 30 40-1000
-500
0
500
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-1000
-500
0
500
velocity, in/s
Forc
e, lb
s
Figure 3.16. LPF1 model, force-velocity plot
0 1 2 3 4 5 6 7 8 9-1000
-500
0
500
time, seconds
Forc
e, lb
s
ModelDataError
0 1 2 3 4 5 6 7 8 9-1000
-500
0
500
time, seconds
Forc
e, lb
s
ModelDataError
Figure 3.17. LPF1 model, error vs. time
58
3.3.3 LPNL2 Model
The LPNL2 model blends the force predicted by the Polylinear model with the force from
the LPF1 model. At low velocities, the force is determined by the LPF1 model. At
higher velocities, the force is determined by the Polylinear model. Between the low and
high velocity regions, the force is a linear combination of the Polylinear and LPF1
models.
The time constant and the high and low velocity transition forces that define the LPNL2
model were found using MATLAB’s constrained optimization function fmincon. The
time constant was bounded to be from 0.1 to 10 milliseconds. The high and low velocity
transition forces were bounded between 10 and 1000 lb. They were also constrained so
that the low velocity transition force occurs at a lower force than the high velocity
transition force with a minimum difference of 1 lb to avoid a large slope in the weighting
function.
The fit resulted in a time constant of 1.3 seconds, which agrees with the time constant in
the LPF1 model. The transition forces were 560 lb and 1000 lb. The fit error was 8.0 lb
RMS, which is only slightly smaller than the 8.1 lb RMS error for the LPF1 model. The
reason for this small improvement in error is that the data for this particular shock build
does not exhibit the near-zero hysteresis at high velocities that previous studies exhibit.
While this does not indicate a significant benefit of LPNL2 versus LPF1, further analysis
of the validation data in Section 3.4 will show that LPNL2 does not do worse than LPF1
and performs noticeably better for some shock builds and drivefiles.
3.3.4 Summary
This section has described the fitting of three different types of shock absorber models for
shock build 1 at 50 psi and 1 click on the rebound adjuster. A summary of the results of
the fits is shown in Table 3.4. The fit time and evaluation times are the times required to
fit the model and to evaluate the model for a 9 second velocity input sampled at 2000 Hz
on a 1.8 GHz Pentium M laptop with 512M RAM, and are given for comparison
purposes. While Polylinear must be evaluated as an input to LPF1 and LPNL2, the
evaluation times for LPF1 and LPNL2 do not include the evaluation time for Polylinear.
59
Table 3.4. Summary of model fitting for build 1, 50 psi, 1 click
Fit Error Fit Time Evaluation Timelb RMS s s
Polylinear 29.38 - 8.0% 0.12 0.02LPF1 8.1 72.3% 2.2% 32 0.03LPNL2 8.0 72.7% 2.2% 181 0.04
% of Signal STD
% Reduction from Static
The Polylinear curve fit model does a good job modeling the shock, with a RMS error of
29 lb, or 8% of the measured force standard deviation. The fit and evaluation times are
very small.
The dynamic LPF1 model does significantly better than Polylinear, with a 72% reduction
in RMS error. The fit time of 32 seconds could be reduced further by using an
optimization routine with a good initial guess or an exhaustive search over fewer lag
values. Since the differential equations are linear, evaluation can be done quickly using
MATLAB’s lsim function.
The LPNL2 only does slightly better than LPF1. This is because the particular build
tested does not exhibit much of the near-zero lag at high velocities that LPNL2 was
designed to accommodate. Further validation analysis will show some significant
improvements in LPNL2 over LPF1. The fit time of 181 seconds could be greatly
reduced by using a better initial guess of the time constant and the transition forces.
Further improvement could be made if the transition forces were fixed based on
inspection of the force-velocity curves.
The model fitting was repeated for all 12 shock configurations described in Section 3.2.
The effect of shock configuration on modeling error will be explored in the next section.
3.4 Model Validation
This section analyzes how well the different shock models are able to predict the shock’s
performance. The first section will present the results of the random testing. The next
section will present the sine testing, which will allow us to determine which amplitudes
and velocities the model will do well and where it may have problems. The third section
60
presents the static testing, which allows us to determine the effect of the gas spring and
seal drag. The last section presents some bump testing results, which indicates the model
performance for extreme disturbances.
3.4.1 Random Testing
This section presents the validation results of the random testing. The velocity profile
used for random testing was used to predict the shock force using the three different
model types for each of the 12 different builds. The model force was then compared to
the actual force for that particular shock configuration and the RMS error was calculated,
as shown in Table 3.5 and Figure 3.18.
Table 3.5. RMS error analysis, lb
Build 1 Build 1 Build 1 Build 11 click 14 clicks 1 click 14 clicks50 psi 50 psi 150 psi 150 psi
Polylinear 29.05 34.04 27.2 32.7LPF1 7.5 9.2 7.9 9.8LPNL2 7.5 8.7 7.6 9.2
Build 2 Build 2 Build 2 Build 21 click 14 clicks 1 click 14 clicks50 psi 50 psi 150 psi 150 psi
Polylinear 41.5 45.5 22.3 28.1LPF1 22.7 22.6 6.1 7.4LPNL2 21.2 21.5 6 7
Build 4 Build 4 Build 4 Build 41 click 14 clicks 1 click 14 clicks50 psi 50 psi 150 psi 150 psi
Polylinear 17.9 23.2 27.2 24LPF1 5.3 7.8 10.0 8.3LPNL2 5.2 7.7 9.8 8.2
61
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
PolylinearLPF1LPNL2
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
(a)
(b)
(c)
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
PolylinearLPF1LPNL2
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
(a)
(b)
(c)
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
PolylinearLPF1LPNL2
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
(a)
(b)
(c)
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
PolylinearLPF1LPNL2
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
1 click, 50 psi 14 clicks, 50 psi 1 clicks, 150 psi 14 clicks, 150 psi0
10
20
30
40
50
RM
S E
rror,
lbs
(a)
(b)
(c)
Figure 3.18. Random validation: (a) Build 1, (b) Build 2, (c) Build 4
Several significant observations can be made from these results. First note that in all
cases the error for the Polylinear model is much larger than the error for the two dynamic
models. The dynamic models ranged from 45-75% less error than the Polylinear model.
The errors for the two dynamic models are similar, but the error for the LPNL2 model is
less in all cases. The reduction in error from LPF1 to LPNL2 ranged from 1-7%. It is
important to note that all of the builds tested had some hysteresis at the highest velocities,
so the benefits of the LPNL2 model may not be evident here.
The errors were smallest for build 4, which had softer compression shims. This build had
the least amount of hysteresis, with 18 lb RMS error for the Polylinear model at 50 psi
and one click, compared to 29 and 42 lb for the other two builds.
The errors were the largest for build 2 at 50 psi, which had softer head valve compression
shims. This makes sense, as the reason for using a head valve is that lower gas pressures
62
can be run without increased hysteresis. This is due to the large amount of hysteresis in
the rebound stroke in build 2 at low pressure, as seen in Figure 3.19. During the
compression stroke at low pressure, not enough pressure is built up in the compression
chamber to create the required flow to the rebound chamber. This causes fluid expansion
in the rebound chamber, which must be compressed during the rebound stroke before the
force can fully develop. This creates a lag in the force-velocity plot. When we decreased
the head valve compression shims, we increased the fluid expansion in the rebound
chamber during compression, increasing the lag in the rebound stroke. This effect was
eliminated when we increased the gas pressure to 150 psi.
Also notice that the percent improvement for the dynamic models compared with the
algebraic model is worst for build 2, low pressure. The improvement was only about
45% for this case, while most other cases showed 60-75% improvement. This indicates
that the dynamic models do not do as well with large hysteresis.
An important observation is that the largest modeling errors occur in shocks that do not
perform as well as the others. The modeling error is low for shocks with good
performance that we may want to put on the car. This indicates that the dynamic models
are sufficient for simulations to help choose the best setups. If it is necessary to model a
shock with a large pocket of hysteresis like the one in Figure 3.19a, it may be possible to
add another lagged state that only influenced the desired velocity range.
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
(a) (b)
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
(a) (b)
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
(a) (b)
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
(a) (b)
Figure 3.19. Force-velocity for build 2, 1 click: (a) 50 psi, (b) 150 psi
63
For builds 1 and 2, the error increased with a more restrictive bleed setting and lower gas
pressure. Both of these factors generally increase hysteresis, causing more modeling
error.
Based on the trends with the bleed and head valves, one might expect that softening the
head valve compression shims would increase the error. This is true for softening the
head valve with low gas pressure, but the exact opposite trend is seen when softening the
head valve at high pressure. This can be explained by recalling that the head valve is
used to prevent fluid expansion without high gas pressures. At low pressures, where the
head valve is working as it was designed, reducing the effectiveness of the head valve
increases hysteresis and increases modeling error. At high pressures, the head valve is
not needed and acts as another flow restriction, actually increasing compressibility
effects. As the effect of the head valve is reduced, the flow restriction is lessened,
reducing hysteresis and modeling error. The interaction between the head valve and the
gas pressure seem to be a significant factor influencing model accuracy.
3.4.2 Sine Testing
This section presents the sine test validation results, which will give us a better feel for
where the model performs well and where it breaks down. The response for the three
shock model types and the 12 shock configurations was calculated for the 14 sine waves
listed in Table 3.3 and the simulated response was compared to the measured response.
The RMS error for each test was calculated and compiled for analysis.
The first thing we consider is how the different model types compare at different
amplitudes and frequencies. Figure 3.20 shows how the three model types compare for
build 1 with 50 psi gas and 14 clicks rebound. Similar trends exist for the other shock
configurations.
64
1 in/s 3 in/s 6 in/s 10 in/s 20 in/s 30 in/s 5 in/s 10 in/s 20 in/s 10 in/s 20 in/s 2.5 in/s 5 in/s 10 in/s0
10
20
30
40
50
60
RM
S E
rror,
lbs
PolylinearLPF1LPNL2
1 inch 0.5 in 0.1 in0.25 in
0.2 Hz 0.5 Hz 1.0 Hz 1.6 Hz 3.2 Hz 4.8 Hz 1.6 Hz 3.2 Hz 6.4 Hz6.4 Hz 12.7 Hz 4.0 Hz 8.0 Hz 15.9 Hz
1 in/s 3 in/s 6 in/s 10 in/s 20 in/s 30 in/s 5 in/s 10 in/s 20 in/s 10 in/s 20 in/s 2.5 in/s 5 in/s 10 in/s0
10
20
30
40
50
60
RM
S E
rror,
lbs
PolylinearLPF1LPNL2
1 inch 0.5 in 0.1 in0.25 in
0.2 Hz 0.5 Hz 1.0 Hz 1.6 Hz 3.2 Hz 4.8 Hz 1.6 Hz 3.2 Hz 6.4 Hz6.4 Hz 12.7 Hz 4.0 Hz 8.0 Hz 15.9 Hz
1 in/s 3 in/s 6 in/s 10 in/s 20 in/s 30 in/s 5 in/s 10 in/s 20 in/s 10 in/s 20 in/s 2.5 in/s 5 in/s 10 in/s0
10
20
30
40
50
60
RM
S E
rror,
lbs
PolylinearLPF1LPNL2
1 inch 0.5 in 0.1 in0.25 in
0.2 Hz 0.5 Hz 1.0 Hz 1.6 Hz 3.2 Hz 4.8 Hz 1.6 Hz 3.2 Hz 6.4 Hz6.4 Hz 12.7 Hz 4.0 Hz 8.0 Hz 15.9 Hz
1 in/s 3 in/s 6 in/s 10 in/s 20 in/s 30 in/s 5 in/s 10 in/s 20 in/s 10 in/s 20 in/s 2.5 in/s 5 in/s 10 in/s0
10
20
30
40
50
60
RM
S E
rror,
lbs
PolylinearLPF1LPNL2
1 inch 0.5 in 0.1 in0.25 in
0.2 Hz 0.5 Hz 1.0 Hz 1.6 Hz 3.2 Hz 4.8 Hz 1.6 Hz 3.2 Hz 6.4 Hz6.4 Hz 12.7 Hz 4.0 Hz 8.0 Hz 15.9 Hz
Figure 3.20. Comparison of sine validation error versus model type
for build 1, 14 clicks, 50 psi
The first thing to notice is that the error for the dynamic models is below 25 lb RMS for
all tests except for the low velocity, low frequency. This is also the case with all shock
configurations except for the low pressure, soft head valve configuration that was found
to have the largest errors discussed in the random validation due to fluid compressibility.
This shock configuration had errors exceeding 25 lb RMS at 1 inch at 20 in/s and 30 in/s
and at 0.5 inch at 20 in/s. The generally low errors across all shock configurations
indicate that the model is reasonable for simulation applications.
For each amplitude, the errors for the three different model types are similar at the lowest
velocities, but the dynamic models outperform the algebraic model at higher frequencies.
For the 1, 0.5, and 0.1 inch amplitudes, the largest frequencies where the algebraic and
dynamic models have almost identical error are 1.6, 1.6, and 4 Hz, respectively. Above
these frequencies, the improvement in using the dynamic models is significant. This
seems to indicate that the shock dynamics only become significant above roughly 2-4 Hz,
at which point there is enough dynamic behavior that a dynamic model is warranted.
Next, notice that while error grows at higher frequencies, the largest error of 32 lb RMS
actually occurs at the lowest amplitude and frequency. The reason for this can be seen in
Figure 3.21. In Figure 3.21a we see the force-velocity plot at 1 and 6 in/s. Also shown is
the Polylinear model. While the data exhibits a large nose, the model is almost linear.
65
This is because the model was fit over a large velocity range, where the low speed nose is
not as noticeable. At higher velocities or with more hysteresis, the effect of the low
speed nose is not noticeable, as shown in Figure 3.21b. This issue should be considered
before collecting data to make shock models. If a better fit at low velocity and low
frequency is desired, this could be achieved by fitting the low velocity polynomial in the
Polylinear model using more low velocity data.
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
1 inch, 30 in/sPolylinear ModelLPNL2 Model
-6 -4 -2 0 2 4 6-250
-200
-150
-100
-50
0
50
100
150
200
250
velocity, in/s
Forc
e, lb
s
1 inch, 1 in/s1 inch, 6 in/sPolylinear Model
(a) (b)
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
1 inch, 30 in/sPolylinear ModelLPNL2 Model
-6 -4 -2 0 2 4 6-250
-200
-150
-100
-50
0
50
100
150
200
250
velocity, in/s
Forc
e, lb
s
1 inch, 1 in/s1 inch, 6 in/sPolylinear Model
(a) (b)
-6 -4 -2 0 2 4 6-250
-200
-150
-100
-50
0
50
100
150
200
250
velocity, in/s
Forc
e, lb
s
1 inch, 1 in/s1 inch, 6 in/sPolylinear Model
(a) (b)
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
1 inch, 30 in/sPolylinear ModelLPNL2 Model
-6 -4 -2 0 2 4 6-250
-200
-150
-100
-50
0
50
100
150
200
250
velocity, in/s
Forc
e, lb
s
1 inch, 1 in/s1 inch, 6 in/sPolylinear Model
(a) (b)
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
600
800
velocity, in/s
Forc
e, lb
s
1 inch, 30 in/sPolylinear ModelLPNL2 Model
-6 -4 -2 0 2 4 6-250
-200
-150
-100
-50
0
50
100
150
200
250
velocity, in/s
Forc
e, lb
s
1 inch, 1 in/s1 inch, 6 in/sPolylinear Model
(a) (b)
-6 -4 -2 0 2 4 6-250
-200
-150
-100
-50
0
50
100
150
200
250
velocity, in/s
Forc
e, lb
s
1 inch, 1 in/s1 inch, 6 in/sPolylinear Model
(a) (b)
Figure 3.21. Force-velocity for sine tests: (a) low velocity, (b) high velocity
The next thing to notice is that the LPNL2 model is slightly better than the LPF1 at
higher frequencies, where hysteresis is more significant. The reason for this can be seen
in Figure 3.21b. At low velocities, the data exhibits 200 lb of hysteresis, but at higher
velocities, there is very little hysteresis. The transition between LPF1 at low velocity and
Polylinear at high velocity allows the model to have lag at low velocity and quickly
transition to small lag at high velocity.
3.4.3 Static Testing
Static testing was conducted for the 6 combinations of build and gas pressure for the low
bleed setting. The drivefile starts at an initial test offset of 2 inches and stays there for 20
seconds to allow the shock pressures to equalize. It then moves up one inch at 0.2 in/s
and maintains the new position for 20 seconds. This process is repeated until the shock is
moved up 3 inches from the original test offset, at which point the process is repeated in
the downward direction until the shock reaches the initial test offset of 2 inches. The
66
static test gives a spring rating of the shock at three points, which gives us both the force-
deflection curve and also the amount of hysteresis in that curve caused by directional
effects such as seal drag. Since this process eliminates the damping effect, this gives us
an indication of the stiffness in parallel to the damper.
The time histories of the force for the 6 static tests are shown in Figure 3.22. For
reference, the drive profile is also shown in Figure 3.22a. For comparison purposes, each
signal was shifted so the force at zero time was zero. The time ranges where the position
was held constant are clearly marked on Figure 3.22b. The high force regions between
static positions are when it was moving at 0.2 in/s. Once the static position was reached,
the shock took about 5 seconds to equalize when compressing and no time in rebound.
This is because the shock has a rebound bleed with a check valve that is closed in
compression.
0 20 40 60 80 100 120 140 160 180-60
-40
-20
0
20
40
60
80
time, seconds
Sta
tic F
orce
, lbs
build1_01_50build1_01_150build2_01_50build2_01_150build4_01_50build4_01_150
0 in 1 in 2 in 3 in 2 in 1 in 0 in
0 20 40 60 80 100 120 140 160 180-2
0
2
4
time, seconds
Driv
e, in
(a)
(b)
0 20 40 60 80 100 120 140 160 180-60
-40
-20
0
20
40
60
80
time, seconds
Sta
tic F
orce
, lbs
build1_01_50build1_01_150build2_01_50build2_01_150build4_01_50build4_01_150
0 in 1 in 2 in 3 in 2 in 1 in 0 in0 20 40 60 80 100 120 140 160 180
-60
-40
-20
0
20
40
60
80
time, seconds
Sta
tic F
orce
, lbs
build1_01_50build1_01_150build2_01_50build2_01_150build4_01_50build4_01_150
0 in 1 in 2 in 3 in 2 in 1 in 0 in
0 20 40 60 80 100 120 140 160 180-2
0
2
4
time, seconds
Driv
e, in
(a)
(b)
0 20 40 60 80 100 120 140 160 180-60
-40
-20
0
20
40
60
80
time, seconds
Sta
tic F
orce
, lbs
build1_01_50build1_01_150build2_01_50build2_01_150build4_01_50build4_01_150
0 in 1 in 2 in 3 in 2 in 1 in 0 in
0 20 40 60 80 100 120 140 160 180-2
0
2
4
time, seconds
Driv
e, in
(a)
(b)
0 20 40 60 80 100 120 140 160 180-60
-40
-20
0
20
40
60
80
time, seconds
Sta
tic F
orce
, lbs
build1_01_50build1_01_150build2_01_50build2_01_150build4_01_50build4_01_150
0 in 1 in 2 in 3 in 2 in 1 in 0 in0 20 40 60 80 100 120 140 160 180
-60
-40
-20
0
20
40
60
80
time, seconds
Sta
tic F
orce
, lbs
build1_01_50build1_01_150build2_01_50build2_01_150build4_01_50build4_01_150
0 in 1 in 2 in 3 in 2 in 1 in 0 in
0 20 40 60 80 100 120 140 160 180-2
0
2
4
time, seconds
Driv
e, in
(a)
(b)
Figure 3.22. Static test time plots: (a) Drive profile, (b) Force
67
The data from Figure 3.22 was reduced to a force-deflection plot by averaging over the
time held at each static position, excluding the first 5 seconds and last second of each
interval. To allow the results of all 6 force-displacement tests to be considered in one
plot, each plot was shifted so that the force at zero position when moving in compression
was zero for each test. The resulting force-deflection plot is shown in Figure 3.23.
0 0.5 1 1.5 2 2.5 3-10
-5
0
5
10
15
Position, in
Sta
tic F
orce
, lbs
Build 1, 50 psiBuild 1, 150 psiBuild 2, 50 psiBuild 2, 150 psiBuild 4, 50 psiBuild 4, 150 psi
Compression
Rebound
0 0.5 1 1.5 2 2.5 3-10
-5
0
5
10
15
Position, in
Sta
tic F
orce
, lbs
Build 1, 50 psiBuild 1, 150 psiBuild 2, 50 psiBuild 2, 150 psiBuild 4, 50 psiBuild 4, 150 psi
Compression
Rebound
0 0.5 1 1.5 2 2.5 3-10
-5
0
5
10
15
Position, in
Sta
tic F
orce
, lbs
Build 1, 50 psiBuild 1, 150 psiBuild 2, 50 psiBuild 2, 150 psiBuild 4, 50 psiBuild 4, 150 psi
Compression
Rebound
0 0.5 1 1.5 2 2.5 3-10
-5
0
5
10
15
Position, in
Sta
tic F
orce
, lbs
Build 1, 50 psiBuild 1, 150 psiBuild 2, 50 psiBuild 2, 150 psiBuild 4, 50 psiBuild 4, 150 psi
Compression
Rebound
Figure 3.23. Static test force-displacement
This plot shows essentially two different force-deflection curves, one for 50 psi, as
indicated by the thin lines, and a stiffer one for 150 psi, indicated by the thick lines.
Notice however that the increase in force for the 150 psi curve is only about 12 lb over 3
inches, at a rate of 4 lb/in. This is much lower than the primary suspension rate,
indicating that the parallel spring effect is minimal.
Also note that there is about a 6-10 lb offset in the force-deflection curve between
compression and rebound. This is most likely caused by seal drag on the shaft bearing,
main piston, and floating piston.
It is possible that this could be integrated into a shock model to further improve the
accuracy. However, the above analysis shows that these effects are very small in
68
comparison with the forces seen in the shock during operation, so any benefit would be
minimal.
3.4.4 Bump Testing
Each of the three builds were also tested using a 1 inch bump test, with the drivefile
shown in Figure 3.13. The transition speed of the bump was 30 in/s. The purpose of the
bump test is to determine how well the model performs for a transient input.
Figure 3.24 shows the force for a compression bump from measured data and predicted
by Polylinear and LPNL2. The ripple is caused because the dynamometer was not able to
perfectly match our bump profile. As this figure shows, both the Polylinear and LPNL2
models do an excellent job of predicting the bump force in both compression and
rebound. Similar responses were observed for the other shock builds.
Both models do well in predicting the general shape of the response, but LPNL2 does a
slightly better job at matching the phase of the response, as evident in the force-velocity
plot in Figure 3.25. LPNL2 matches the response very well in rebound, but does only
slightly better than Polylinear in compression. These results suggest that the models are
capable of accurately reproducing transient inputs.
5 08 5 1 5 12 5 14 5 16 5 18 5 2
-100
-50
0
50
100
150
200
250
300
Forc
e, lb
s
DataPolylinearLPNL2
70 25 70 26 70 27 70 28 70 29 70 3 70 31 70 32 70 33
-700
-600
-500
-400
-300
-200
-100
0
100
Forc
e, lb
s
DataPolylinearLPNL2
5 08 5 1 5 12 5 14 5 16 5 18 5 2
-100
-50
0
50
100
150
200
250
300
Forc
e, lb
s
DataPolylinearLPNL2
70 25 70 26 70 27 70 28 70 29 70 3 70 31 70 32 70 33
-700
-600
-500
-400
-300
-200
-100
0
100
Forc
e, lb
s
DataPolylinearLPNL2
DataPolylinearLPNL2
DataPolylinearLPNL2
DataPolylinearLPNL2
DataPolylinearLPNL2
5.08 5.1 5.12 5.14 5.16 5.18 5.2 70.25 70.26 70.27 70.28 70.29 70.3 70.31 70.32 70.33time, seconds time, seconds
300
250
200
150
100
50
0
-50
-100
100
0
-100
-200
-300
-400
-500
-600
-700
Forc
e, lb
Forc
e, lb
(a) (b)
5 08 5 1 5 12 5 14 5 16 5 18 5 2
-100
-50
0
50
100
150
200
250
300
Forc
e, lb
s
DataPolylinearLPNL2
70 25 70 26 70 27 70 28 70 29 70 3 70 31 70 32 70 33
-700
-600
-500
-400
-300
-200
-100
0
100
Forc
e, lb
s
DataPolylinearLPNL2
5 08 5 1 5 12 5 14 5 16 5 18 5 2
-100
-50
0
50
100
150
200
250
300
Forc
e, lb
s
DataPolylinearLPNL2
70 25 70 26 70 27 70 28 70 29 70 3 70 31 70 32 70 33
-700
-600
-500
-400
-300
-200
-100
0
100
Forc
e, lb
s
DataPolylinearLPNL2
DataPolylinearLPNL2
DataPolylinearLPNL2
DataPolylinearLPNL2
DataPolylinearLPNL2
5.08 5.1 5.12 5.14 5.16 5.18 5.2 70.25 70.26 70.27 70.28 70.29 70.3 70.31 70.32 70.33time, seconds time, seconds
300
250
200
150
100
50
0
-50
-100
100
0
-100
-200
-300
-400
-500
-600
-700
Forc
e, lb
Forc
e, lb
(a) (b)
Figure 3.24. Force vs. time for build 4 bump test: (a) Compression, (b) Rebound
69
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
velocity, in/s
Forc
e, lb
s
DataPolylinearLPNL2
-40 -30 -20 -10 0 10 20 30 40-800
-600
-400
-200
0
200
400
velocity, in/s
Forc
e, lb
s
DataPolylinearLPNL2
Figure 3.25. Force-velocity plot for build 4 bump test
3.5 Summary
This chapter has developed models for a Penske 7300 shock absorber, which is used in
NASCAR Cup and similar racing series. These models can be created using simple
shock dynamometer tests and then be integrated in a vehicle simulation.
First, we presented a simplified physical model of a shock absorber with compressible
flow. The simplified physical model was then used to motivate models that could be fit
to real shock data. The first model (Polylinear) was a simple curve fit, based on the fact
that most of the shock absorber’s response is a function of velocity alone.
While the algebraic model can predict the general trend of the force-velocity curve, it
cannot characterize the hysteresis that is significant in many shock curves. Motivated by
our mechanical equivalent for our compressible model, our first dynamic model (LPF1)
was a simple first-order lag of the force calculated by the curve fit.
Many force-velocity curves exhibit significant hysteresis at low velocity but almost no
thickness at higher velocities. To accommodate this behavior, a model (LPNL2) was
created to behave similarly to a stiffening spring in series with the damper, by using only
the lagged force at low velocity, only the curve fit at high velocity and a linear
combination of the two models between low and high velocity regions.
70
Three different builds in a total of 12 different configurations were tested on a shock
dynamometer. The drivefiles included random, sine wave, static, and bump. This testing
was designed to allow us to investigate the ability of the models to characterize the shock
absorbers for different settings under different operating conditions. The random data
was then used to fit models for each of the 12 shock configurations and 3 model types.
After the models were fitted to the experimental data, the models were validated using
another random signal. By comparing the RMS modeling error between different shock
configurations, several important conclusions were made. First, the reduction in error for
the dynamic models compared to the algebraic models was significant, ranging from 45%
to 75%. Aside from the builds with cavitation, RMS errors were low, suggesting the
model is suitable for simulation applications.
The sine wave validation shows that both the algebraic and dynamic modeling errors
were very similar below 2 Hz, suggesting that dynamic models do not provide much
value below 2 Hz. Larger errors were also present at the lowest frequencies, because the
broadband fit data placed less emphasis on the low velocity nose only significant at lower
frequency. This could potentially be improved by fitting the low velocity curve fit using
low frequency data. The RMS errors were generally low, being below 25 lb RMS in
most cases.
Static validation showed that the gas spring stiffness contributed about 10 lb at 3 inches
for the 150 psi case, and showed about 5 lb of friction. The bump validation showed that
the shock models could accurately predict transient events.
Now that we have developed validated shock models, we can proceed to integrate them
into vehicle simulations. We begin by applying these models in a quarter-car vehicle
model in Chapter 4.
71
Chapter 4 Quarter-Car Model Development
During 8-post rig testing, a vehicle is tested by repeating a particular drivefile as various
suspension settings are tuned. The performance of the different runs is compared by
calculating a series of performance metrics. These metrics are typically used during 8-
post rig tests to help understand how the shock selection affects mechanical grip,
aerodynamics, and other factors.
This sensitivity information is one of the main products of the 8-post rig test. The race
engineers will use this information to help select or eliminate potential shock packages to
try on the race track, and will use the sensitivity information to help make adjustments
based on how the vehicle performs on the track.
The main goal of this research is to accurately predict this sensitivity information using
simulations, which can augment knowledge gained from 8-post rig testing. To create an
accurate model for these simulations, we will use rig test data to perform system
identification, which will produce a vehicle model. To make sure our system
identification methods are working properly, we will first run the ID on simulated data
with a known model. This will allow us to compare the true model with the ID model
and tune our ID methods.
Once the ID methods have been developed by performing ID on simulated data, the ID
method is then applied to data collected from actual rig data. The ability of the ID model
to predict the influence of shock selection on the quarter-car rig’s response is then
evaluated.
This chapter develops the methods at a quarter-car scale. The first section develops the
equations of motion for a linear quarter-car model. This model treats the nonlinear shock
model force as an input to a linear dynamic system. This allows us to decouple the
nonlinear shock model from the linear vehicle model and consider the frequency response
72
of the linear vehicle model. A parametric analysis of the linear vehicle model is
performed to provide a better understanding of the effect of shock selection on vehicle
performance.
Once we have a basic understanding of the behavior of the linear vehicle model, we then
couple the linear dynamic model with the nonlinear shock model. This model is used in
combination with the shock models presented in Chapter 3 to generate simulated quarter-
car rig data.
The methods required to perform the system ID are then discussed and applied to the
simulated test data. The results of the ID are compared with the theoretical response to
verify that the ID process is working properly.
To verify that the method will work on real-world data, the ID process is then applied to
real quarter-car rig data and the ID model is used to predict the response for different
shock configurations.
4.1 Quarter-Car Model, Decoupled Analysis
The section develops the equations of motion for a quarter-car vehicle model, where the
nonlinear shock model force is treated as an input to a linear dynamic vehicle model.
This allows us to decouple the nonlinear shock model from the linear vehicle model and
apply linear analysis to better understand the effect of shock selection on vehicle
performance.
This section develops the decoupled quarter-car model with a shock force input. Once
the model is developed, the behavior of the linear model is analyzed using a parametric
study of the frequency response.
4.1.1 State-Space Model
A diagram of the quarter-car model is shown in Figure 4.1. It includes two masses,
which represent the inertia of the chassis and the wheel assembly. The two masses are
73
connected through the suspension, represented by a linear spring and damper. The linear
suspension parameters represent wheel rates that take into account the motion ratio
between the wheel and the physical spring or damper. The linear damping rate can be
used to represent a linear shock absorber or other losses in the suspension.
m1
m2
k c Fs
kt
w1
w2
yR
FA
m1
m2
k c Fs
kt
w1
w2
yR
FA
Figure 4.1. The quarter-car model
The suspension model also includes a generic force input, which will be used in Section
4.2 to integrate our nonlinear shock absorber models. Note however that this is a generic
force input that acts on both the sprung and unsprung masses, so it could also be used to
add other types of suspension behavior, such as progressive springs, coil-binding, and
suspension friction.
The tire is represented as a linear spring. In addition to the shock force input, there are
other inputs that represent the road profile and aeroloader force. During rig testing, the
road inputs are simulated using wheelloaders and the aerodynamic inputs are simulated
using aeroloaders.
The equation of motion for the quarter-car model in Figure 4.1 is
1 1 1 1
2 2 2 2
0 1 100 10
R
At t
S
yk km w w wc c
Fk k k km w w wc c
FMw Cw Kw Lu
− −− + + = − + −−
+ + =
(4.1)
74
where the shock force is positive during compression. Rewriting Equation (4.1) in state-
space form yields
2 2 2 2 31 1 1
0 0
x xw I wu
w M K M C w M Lx Ax Bu
− − −
= + − − = +
(4.2)
The output equations are chosen to represent common sensor measurements. From an
identification standpoint, it can be important if a sensor is a function of the state alone, a
function of only the state and the inputs (direct passthrough), or a function of the states,
inputs and system parameters. The sensor equations for the position and velocity of the
chassis, wheel, and suspension are a function of the state alone
1
2
1
2
1 0 0 00 1 0 00 0 1 00 0 0 11 1 0 0
0 0 1 1s
s
www
xwxx
= −
−
(4.3)
The ride height is a function of both the state and the road input
[ ] [ ]1 0 0 0 1 0 0RHx x u= + − (4.4)
The spring force, tire force, chassis accelerometer, and wheel accelerometer sensor
equations include the states, inputs, and system parameters.
( )( )
( )( )
1
2
0 0 0 0 00 0 0 0 0
3,: 3,:4,: 4,:
spring
t tt
k kFk kF
x uA BwA Bw
− − = +
(4.5)
where A(i,:) and B(i,:) refers to the MATLAB notation for the ith row of matrices A and
B. Combining sensor Equations (4.3)-(4.5) and rearranging sensor order yields
75
( )( )
( )( )
1
2
1
2
1
2
1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 01 1 0 0 0 0 0
0 0 1 1 0 0 03,: 3,:4,: 4,:
0 0 0 0 01 0 0 0 1 0 0
0 0 0 0 0
s
s
t t t
RH
spring
wwwwxx xw A Bw A BF k k
xF k k
−
= +− − − −
u
y Cx Du
= + (4.6)
The decoupled quarter-car model is then found in state-space form by combining
Equations (4.2) and (4.6) to get
x Ax Buy Cx Du= += +
(4.7)
where the state-space matrices are defined in Equations (4.1), (4.2), and (4.6).
Now that we have formed the linear model, we can study the behavior of the system by
analyzing the effect of different system parameters on the model’s frequency response.
For the purpose of this analysis, a set of baseline parameters was chosen to be generally
representative of suspensions used in the racing industry, while not representing any
specific suspension. The parametric study will consider the parametric sensitivity around
this baseline setup.
The baseline parameter values are shown in Table 4.1. The sprung and unsprung masses
correspond to a 3000 lb car with 90% sprung mass, and 25% of the mass on one corner.
The suspension and tire stiffnesses are typical values. Notice that unlike a passenger car,
where the tire may be as much as an order of magnitude stiffer than the soft primary
suspension, the stiff suspensions used in racing are often similar to the tire stiffness.
76
Table 4.1. Baseline parameter values
m1 g Sprung Weight 675 lbfm2 g Unsprung Weight 75 lbf
k Suspension Stiffness 1280 lbf/inkt Tire Stiffness 1300 lbf/inc Suspension Damping 16.67 lbf / (in/s)
4.1.2 Frequency Response
The two commonly considered measurements in rig testing are tire normal force variation
and ride height variation, which are indicators of mechanical grip and aero grip. This
section considers how the frequency response for the tire force and ride height changes
with variations in suspension parameters. Having an understanding of this fundamental
behavior can be a useful tool when trying to analyze the more complex nonlinear system.
To plot the wide range of FRF magnitude, we will plot magnitude in decibels (dB), which
is defined by
1020 logdB
H H= (4.8)
First, the effect of damping is considered by varying damping from 50-300% of the
nominal value. Figure 4.2 shows how the frequency responses for the tire force vary with
damping. For the road input, the effect of damping is very small until about 2 Hz. Near
the resonance at 3 Hz, damping reduces the resonant peak, reducing tire force variation.
After the resonant peak, damping worsens the attenuation, increasing tire force variation.
The same trend exists for the aeroloader input, but the reduced attenuation at higher
frequencies is less pronounced.
100 10130
40
50
60
70
80
90
Tire
For
ce /
Whe
ello
ader
Pos
ition
, dB
Frequency, Hz
100 101-30
-20
-10
0
10
20
30
Tire
For
ce /
Aer
oloa
der F
orce
, dB
Frequency, Hz100 101
-30
-20
-10
0
10
20
Tire
For
ce /
Sho
ck F
orce
, dB
Frequency, Hz
0.5 * c1 * c2 * c3 * c
Figure 4.2. Effect of damping on tire force
77
The shock force input shows a reduction at all frequencies for increased damping, which
may seem odd at first glance. We can think of the linear spring and damping forces as
the nominal linear suspension forces. The “shock force” can include any additional
suspension forces in addition to the nominal forces, including shock or spring
nonlinearities, friction, or suspension component dynamics. The third subplot in Figure
4.2 can then be thought of as the control authority of the shock force on the tire force.
Increased damping reduces the control authority of the shock force near the resonances,
while it has little effect elsewhere.
Next, we consider the effect of damping on the ride height, as shown in Figure 4.3.
Increasing damping significantly reduces the ride height variation near the resonance for
both the wheelloader and aeroloader inputs. The increased response after the resonance
is small for the parameter values shown. This suggests using higher damping forces to
minimize ride height variations and improve aerodynamic performance. This will also
reduce the tire normal force variation near resonance. The tradeoff is that increasing
damping will also worsen the attenuation in tire normal force from 4-13 Hz. Depending
on the roughness of the track and the aerodynamic sensitivity, a compromise between the
two must be found.
100 101-20
-10
0
10
20
30
Rid
e H
eigh
t / W
heel
load
er P
ositi
on, d
B
Frequency, Hz
100 101-90
-80
-70
-60
-50
-40
-30
Rid
e H
eigh
t / A
erol
oade
r For
ce, d
B
Frequency, Hz100 101
-120
-100
-80
-60
-40
Rid
e H
eigh
t / S
hock
For
ce, d
B
Frequency, Hz
0.5 * c1 * c2 * c3 * c
Figure 4.3. Effect of damping on ride height
The next parameter we consider is the suspension stiffness. The effect of suspension
stiffness on tire force as suspension stiffness is varied from 5-300% of the baseline value
is shown in Figure 4.4. The general trend is that increasing stiffness increases the natural
frequency and decreases the damping ratio. The significant change in damping ratio with
spring stiffness highlights the significance of matching stiffness with damping. This
effect reduces normal force variation before the natural frequency and increases it after
78
the natural frequency. Also worth noting is the effect of making the suspension very soft.
This effectively decouples the two-mass system into two single-mass systems. This
provides very low tire normal force variations. Also note that softer springs generally
increase the control authority of the shock force, except near resonance.
100
101
30
40
50
60
70
80
90
100
Tire
For
ce /
Whe
ello
ader
Pos
ition
, dB
Frequency, Hz
100
101
-30
-20
-10
0
10
20
30
40
Tire
For
ce /
Aer
oloa
der F
orce
, dB
Frequency, Hz10
010
1-40
-30
-20
-10
0
10
20
30
Tire
For
ce /
Sho
ck F
orce
, dB
Frequency, Hz
0.05 * k1 * k2 * k3 * k
Figure 4.4. Effect of suspension stiffness on tire force
Next, we consider the effect of suspension stiffness on ride height, shown in Figure 4.5.
Similar to the tire force, increasing stiffness increases natural frequency and decreases
damping ratio. This produces a significant reduction in ride height variation at lower
frequencies. Again, there is a tradeoff between mechanical grip and aerodynamic
performance that needs to be considered to select the best combination for a particular
track.
100
101
-30
-20
-10
0
10
20
30
40
Rid
e H
eigh
t / W
heel
load
er P
ositi
on, d
B
Frequency, Hz
100
101
-90
-80
-70
-60
-50
-40
-30
-20
Rid
e H
eigh
t / A
erol
oade
r For
ce, d
B
Frequency, Hz10
010
1-120
-100
-80
-60
-40
Rid
e H
eigh
t / S
hock
For
ce, d
B
Frequency, Hz
0.05 * k1 * k2 * k3 * k
Figure 4.5. Effect of suspension stiffness on ride height
This section has developed a decoupled linear quarter-car model and explored the
influence of changing suspension parameters on the frequency response of the tire normal
force and ride height. A tradeoff has been shown, which generally favors more coupling
between the chassis and wheel to reduce ride height variation and less coupling to reduce
79
normal force variation. The importance of selecting the right combination of shock and
spring was also highlighted.
Now that we have developed the linear quarter-car model and have a basic understanding
of how it performs, we can now integrate the nonlinear shock models created in Chapter
3 into the simulation.
4.2 Quarter-Car Model, Simulated Data
Now that we have defined shock absorber models and a decoupled linear vehicle model,
we can combine the two to create a complete quarter-car vehicle model, as shown in
Figure 4.6. This model can then be used to create simulated data sets, which we will use
in our identification process to determine if our ID was successful in reproducing the
known model.
Linear Vehicle Model
Vehicle Response
Nonlinear Dynamic
Shock Model
Shock Velocity
Shock Force
Road
Aerodynamics Linear Vehicle Model
Vehicle Response
Nonlinear Dynamic
Shock Model
Nonlinear Dynamic
Shock Model
Shock Velocity
Shock Force
Road
Aerodynamics
Figure 4.6. Coupling of vehicle and shock absorber models
The quarter-car model equations with inputs for road displacement, aerodynamic force,
and shock force were derived in Section 4.1. To couple the shock model with the
quarter-car model, the shock velocity from the quarter-car model is treated as an input to
the shock model, and the shock force is treated as an input to the quarter-car model.
The shock model presented in Chapter 3 included a curve fit of the force-velocity curve, a
first-order lowpass filter of the curve fit, and a weighting function to combine the two.
The equations for these three parts are
( )polyF f v= (4.9)
80
11lag polyF F
sτ=
+ (4.10)
( ) ( )( )2 1LPNL poly lag poly polyF F F F Fα α= + − (4.11)
To improve simulation speed for the linear portions of the simulation, the fourth-order
state-space model in Equation (4.7) and the first-order lowpass filter in Equation (4.10)
are transformed into linear difference equations using the bilinear transformation
1
1
2 11
zsT z
−
−
−= +
(4.12)
Where s is the continuous-time Laplace operator, z-1 is the discrete-time unit delay
operator, and T is the sample time. This transformation maps the entire s jω= axis to
the unit circle, which avoids aliasing and preserves the stability of poles, but causes some
warping in the frequency response at higher frequencies [37]. This transformation is
done using the MATLAB function c2d, with a sample rate of 500 Hz. After transforming
Equations (4.7) and (4.10) into discrete time, they may be written as
[ ] [ ] [ ][ ] [ ] [ ]
1
d d
d d
x k A x k B u k
y k C x k D u k
+ = +
= + (4.13)
( ) ( )1
21
12 21 1
LPNL polylinearzF z F z
zT Tτ τ
−
−
+ = + + −
(4.14)
The final quarter-car model can be implemented in Simulink, as shown in Figure 4.7.
The quarter-car block corresponds to Equation (4.13), while the shock model consists of
the blocks polylinear, first-order lag, and LPNL2 weighting.
81
[Ut' U(:,1)]
y_road
U U(E)
selectshock velocity
y(n)=Cx(n)+Du(n)x(n+1)=Ax(n)+Bu(n)
quarter-car
Ysim
To Workspace
MATLABFunction
PolylinearModel
MATLABFunction
LPNL2 Weighting
bbd(z)
aad(z)First-Order Lag
[Ut' U(:,2)]
F_aero
u_linear
u_linear v
F_poly
F_poly F_lag
F_LPNL2
F_LPNL2
y _linear
Figure 4.7. Simulink layout for quarter-car model
Now that we have assembled the full quarter-car model, we can use it to create simulated
data sets to test our system identification methods. A random drivefile was created for
the road and aerodynamic inputs using the multisine method described in Chapter 3. For
both inputs, a flat spectrum was chosen up to 50 Hz with spectral resolution of 0.01 Hz
and duration 100 seconds. The road input had a span of 2 inches and was zero mean,
while the aerodynamic input had a span of 1000 lb and a mean of 500 lb. This drivefile
was used along with the shock models to create a simulated data set.
4.3 System Identification Methods
Now that we have a simulated data set from a known system, we can apply our system
identification methods and see if the identified model accurately reproduces the system
response. This section describes the system identification methods.
We wish to identify a model to describe the relationship between the inputs from the
road, aerodynamics, and shock force and the measured sensor outputs. This will allow us
to create a model of the complete vehicle and simulate the response for different shock
selections. We have measurements of the inputs and outputs from a test conducted where
the first two inputs were defined independently, but the shock force was defined by the
shock velocity. We will show that the dependence of the shock force on the other two
input causes an issue that should be addressed during the identification process. The
subsystem we seek to identify is shown in the dashed box with respect to the complete
82
system in Figure 4.8. The three inputs u1, u2, and u3 correspond to the wheelloader,
aeroloader, and shock force inputs respectively.
H1Nonlinear Dynamic
Shock Model
u1
H2
H3
+u2
u3
y1
y2
y3
y S vshock Fshock
Linear System to Identify
H1H1Nonlinear Dynamic
Shock Model
Nonlinear Dynamic
Shock Model
u1
H2H2
H3H3
++u2
u3
y1
y2
y3
y SS vshock Fshock
Linear System to Identify
Figure 4.8. Diagram of complete system and the subsystem to identify
First, we will provide a summary of a nonparametric frequency response estimation
method that can be applied for independent inputs. Next, the issue of the shock force
being dependent to the first two inputs will be addressed. Finally, a method to convert
the nonparametric frequency response estimates into a parametric model that is useful for
simulation is discussed.
4.3.1 Nonparametric Frequency Response Estimation
If the inputs were independent, a common practice is to create a nonparametric estimate
of the transfer function matrix, which has a coefficient for each input, output, and
frequency. For simplicity, we show the method for only one frequency and one output.
The model that we want to fit is
1 1 2 2 3 3y H u H u H u= + + (4.15)
where the inputs and outputs consist of the frequency domain component for one
frequency. To fit the model, we will collect several data blocks and choose the model
coefficients to minimize model error between the data blocks. Data blocks can be taken
from a series of tests or from selecting multiple blocks out of a longer data set. The data
for the kth data block will have some deviation from the ideal model. If we assume the
error is caused by the outputs, the error model for the kth data block can be represented as
83
[ ] [ ] [ ] [ ] [ ]1 1 2 2 3 3y k H u k H u k H u k kε= + + + (4.16)
If we collect K data sets, we can arrange Equation (4.15) in matrix form as
[ ]
[ ]
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ]
[ ]
1 2 3 1
2
1 2 3 3
1 1 1 1 1
y u u u HH
y K u K u K u K H K
Y UH
ε
ε
ε
= +
= +
(4.17)
We can choose our nonparametric estimate to minimize the norm of the error vector
2 *J ε ε ε= = (4.18)
This is a linear least-squares regression problem, which yields the solution
( ) ( )1* *
1uu uy
H U U U Y
S S
−
−
=
= (4.19)
where Suu and Suy are the autospectrum and cross-spectrum estimates for the K blocks.
Note that the rank of the autospectrum matrix is at most equal to the rank of U, which
requires that there be at least three linearly independent data block rows and three
independent input columns for a unique solution H to exist.
This process is repeated for all frequencies and outputs to complete the nonparametric
frequency response estimation. The model generated by this nonparametric frequency
response estimation process is often called a H1 frequency response estimate.
4.3.2 Frequency Response Estimation with Dependent Data
The H1 frequency response estimation method described in the previous section requires
three independent inputs for a unique solution of Equation (4.19) to exist. To see why
this condition may be violated in our case, consider a linear shock model
shockF cvcSy
==
(4.20)
84
where the vector S selects the shock velocity from the output vector. Substituting into
Equation (4.15) and solving for y yields
( )( ) [ ]
1 1 2 2 3
3 1 1 2 2
13 1 1 2 2
1 1 2 2
y H u H u H cSyI H cS y H u H u
y I H cS H u H u
y H u H u
−
= + +
− = +
= − +
= +
(4.21)
This means the shock force will be
1 1 2 2
1 1 2 2
shock
shock
F cSy
cS H u H u
F H u H u
=
= +
= +
(4.22)
The shock force is linearly dependent with the other two inputs if the shock force can be
written
1 1 2 2shockF c u c u= + (4.23)
for some complex constants c1 and c2. This proves that it is not possible to find a unique
H1 frequency response estimate when the shock is linear. If the shock has significant
nonlinearity, this will add uncorrelated energy into the shock force input, making a
unique solution possible at this frequency.
One method to address the linear dependence issue is to test the car first with a linear
shock (or no shock), then with a different shock. The first test is used to identify the
transfer functions for the first two inputs only. The second test is used to identify the
transfer function for the shock input only. This two-part system identification is shown
in Figure 4.9. This strategy is motivated by the fact that some systems lose identifiably if
tested while closed-loop feedback is applied, but can be made identifiable by using two
or more different control policies [38, 39]. In our case, the control policy corresponds to
the shock model.
85
H1u1
H2 +u2
y1
y2 y
~
~
ID Experiment 1
(a)
(b)
u3
H1u1
H2
H3
+u2
y1
y2
y3
y Svshock
~
ID Experiment 2
~
Nonlinear Dynamic Shock Model
c
-+
Fshock
Flinear
H1u1
H2 +u2
y1
y2 y
~
~
ID Experiment 1
H1H1u1
H2H2 ++u2
y1
y2 y
~
~
ID Experiment 1
(a)
(b)
u3
H1u1
H2
H3
+u2
y1
y2
y3
y Svshock
~
ID Experiment 2
~
Nonlinear Dynamic Shock Model
c
-+
Fshock
Flinearu3
H1u1
H2
H3
+u2
y1
y2
y3
y Svshock
~
ID Experiment 2
~
Nonlinear Dynamic Shock Model
c
-+
Fshock
Flinear
H1H1u1
H2H2
H3H3
++u2
y1
y2
y3
y SSvshock
~
ID Experiment 2
~
Nonlinear Dynamic Shock Model
Nonlinear Dynamic Shock Model
cc
-+-+
Fshock
Flinear
Figure 4.9. System ID experiments: (a) Linear shock for road and aero inputs,
(b) Second shock for shock input
Equation (4.22) shows that the 3 input model of Equation (4.15) can be reduced to two
inputs in the case of a linear shock. The frequency responses 1H and 2H describe the
linear system behavior with the shock included. Note that if the shock is removed, the
frequency responses reduce to 1H and 2H , the original frequency responses in Equation
(4.15). If we run test with a linear shock (or no shock), we can perform the H1
identification procedure to identify 1H and 2H .
Next, we run a second ID experiment with an arbitrary shock. Since the shock force that
would be created by the linear shock from ID experiment 1 is already included in the
frequency response for the first two inputs, we subtract it from the shock force. The
difference between these two forces is then used for input 3.
3 shocku F cv= − (4.24)
86
The shock for the first and second experiment must be different, so there will be energy
in u3. The output due to u3, labeled y3 in Figure 4.8b, can be found since we know 1H
and 2H . In general,
( )
1 2 3
1 1 2 2 3
3 1 1 2 2
y y y y
H u H u y
y y H u H u
= + +
= + +
= − +
(4.25)
If the drivefile for experiment 1 and experiment 2 are the same, this reduces to
3 experiment 1y y y= − (4.26)
where experiment 1y is the output from experiment 1.
Now that we have u3 and y3, we can find the FRF estimate of H3. Combining our
estimate of 1H and 2H from ID experiment 1 and our estimate of H3 from ID experiment
2 gives us a complete nonparametric frequency response estimate of the linear system in
Equation (4.15). Now that we have addressed the issue with the dependence of the shock
force on the other two inputs, we can treat the frequency response estimates as if they
were obtained using a test with three independent inputs.
4.3.3 Parametric System Identification
The nonparametric frequency response estimate consists of as many coefficients as there
are data points in a single block of data. While this could be used for simulation, it is
more practical to fit the nonparametric model with a parametric model with fewer
parameters. The parametric model has the advantage of smoothing the frequency
response function and improving the model in small frequency ranges where the
nonparametric model was poor. It also allows us to see the poles and zeros for the
different input/output pairs.
87
There are many different methods of performing parametric system identification, which
can be chosen depending on the particular application. The transfer function for each
input/output pair can be defined by its poles, zeros, and constant gain. In an ideal linear
system, all input/output pairs should share the same poles and only differ by the zeros and
constant gain. The identification of the system poles is often important for applications
in controls and structural parameter identification.
In our case, it is not necessary to identify a consistent set of system poles. All we want is
to simplify our nonparametric frequency response estimates into a convenient transfer
function relationship that will allow us to simulate the linear system more efficiently.
This allows us to use a simpler method of fitting a single-input single-output (SISO)
transfer function to the frequency response estimate for each input/output pair. This
method will give a different set of poles, zeros, and a constant for each input/output pair.
One advantage of this method is that it is very easy to fit a SISO transfer function, and
many methods are readily available.
Another advantage is that it localizes error in the frequency response estimate to the
corresponding transfer function estimate. If there is significant error in the frequency
response estimate for one of the input/output pairs due to factors such as noise, poor
excitation, nonlinearity, or sensor malfunction, the error will only affect the
corresponding transfer function estimate, not any other transfer function estimate.
The SISO method chosen for this work uses the invfreqs function in MATLAB [40].
This method fits a SISO transfer function of the form
( ) ( )( )
11 2 0
11 2 0
n n
m m
B s b s b s bH s
A s a s a s a
−
−
+ + += =
+ + + (4.27)
by minimizing the equation-error cost function
( ) ( ) ( )( ) ( )( ) 2 2 2 t
kJ w k h k A j f k B j f kπ π= −∑ (4.28)
88
where ( )f k is the frequency at index k, ( )h k is the nonparametric frequency response
estimate, and ( )tw k is a weighting factor. The problem can be solved easily using least
squares regression.
An alternate optimization problem that is more intuitive and often provides better results
is to minimize the output-error cost function
( ) ( )( )( )( )( )
22
2t
k
B j f kJ w k h k
A j f k
π
π= −∑ (4.29)
Unfortunately, this optimization problem is nonlinear in nature and must be solved using
more generic iterative nonlinear programming techniques. The invfreqs function uses a
Gauss-Newton iterative search. The optimal solution to Equation (4.28) is used as an
initial guess for an iterative solution.
4.4 System Identification on Simulated Data
Now that we have developed the required system identification methods, we can then
apply them to identify the linear portion of the quarter-car model from our simulated
data.
4.4.1 System Identification Results
Two simulated ID experiments were performed for use in the system identification
process. Both used the 100-second random signal with a flat spectrum up to 50 Hz
described in Section 4.2. The first ID experiment used a linear shock with a damping
coefficient of 16.6 lb/(in/s). The second ID experiment used the build 1 shock model
with 50 psi gas pressure and 1 click bleed.
The first ID experiment was used to estimate the frequency response 1H and 2H . These
frequency responses correspond to the effect of the road and aerodynamic inputs on the
quarter-car with a linear shock. Twenty data blocks were created from the 100 second
data file using a Hamming window and 50% window overlap. This created data blocks
89
with 9523 samples, corresponding to a 0.105 Hz spectral resolution in the frequency
response estimate.
The second ID experiment was then used to estimate the frequency response H3 using the
same settings as the first experiment. This frequency response determines the influence
of the portion of the shock force in addition to the force that would be created by the
linear shock used in the first ID experiment. The change in shock force input and change
in output measurements for the H1 estimation are calculated using Equations (4.24) and
(4.26).
Once the frequency responses 1H , 2H , and H3 have been estimated, a parametric model
can be fitted to each input/output pair using the invfreqs function. Since we know our
system is 4th order and some sensor equations have direct passthrough, we set both the
numerator and denominator polynomial orders to 4. We also use a weighting function of
1 below 20 Hz and 0 above 20 Hz. This weighting function could be further adjusted if a
frequency response estimate was poor in a particular area or more accuracy was desired
in a particular frequency range.
The results for the nonparametric and parametric ID are shown in Figure 4.10 and 4.11
for the tire force and ride height. Each plot shows the ideal response of the linear system
with the linear shock from ID experiment 1, the H1 frequency response estimate, and the
parametric model fit. As these figures illustrate, both the H1 estimate and the parametric
estimate closely match the ideal.
0 5 10 15 20-20
0
20
40
60
80
Frequency, Hz
Tire
For
ce /
Whe
ello
ader
Pos
ition
, dB
IdealFRF EstimateTF Fit
0 5 10 15 20-30
-20
-10
0
10
20
Tire
For
ce /
Aer
oloa
der F
orce
, dB
Frequency, Hz0 5 10 15 20
-80
-60
-40
-20
0
20
Tire
For
ce /
Sho
ck F
orce
, dB
Frequency, Hz
Figure 4.10. ID results for tire force
90
0 5 10 15 20-30
-20
-10
0
10
20
Frequency, Hz
Rid
e H
eigh
t / W
heel
load
er P
ositi
on, d
B
IdealFRF EstimateTF Fit
0 5 10 15 20-90
-80
-70
-60
-50
-40
-30
Rid
e H
eigh
t / A
erol
oade
r For
ce, d
B
Frequency, Hz0 5 10 15 20
-120
-100
-80
-60
-40
Rid
e H
eigh
t / S
hock
For
ce, d
B
Frequency, Hz
Figure 4.11. ID results ride height
A comparison of the natural frequencies and damping ratios for the transfer function
between road input and chassis position is shown in Table 4.2. This shows good
agreement between the ideal and ID model.
Table 4.2. Comparison of natural frequency and damping ratios
Ideal ID Model Ideal ID Model1 3.04 3.04 6.10 6.382 18.44 18.55 42.19 41.54
Natural Frequency, Hz Damping Ratio, %Mode
4.4.2 Simulation using the ID model
Now that we have identified the linear dynamics of our simulated quarter-car data, we
can now use the ID model for simulation of the effect of changing shock absorbers. This
simulation is almost identical to the simulation that was used to generate the simulated
data, with two exceptions. First, the known linear dynamic model is replaced by the ID
model. Second, the linear damping force from the first ID experiment must be subtracted
from the shock model force. The ID model is discretized using MATLAB’s c2d function
with a sample rate of 500 Hz. The Simulink layout for simulation using the ID model is
shown in Figure 4.12.
91
[Ut' U(:,1)]
y_road
U U(E)
selectshock velocity
qcar.c_bar * 500/30
c_baseline * v
Ysim
To Workspace
MATLABFunction
Polyl inear Model
y(n)=Cx(n)+Du(n)x(n+1)=Ax(n)+Bu(n)
Linear Discrete
MATLABFunction
LPNL2 Weighting[Ut' U(:,2)]
F_aero
bbd(z)
aad(z)Discrete Filter
u_linear v F_poly F_lagy _linear
Figure 4.12. Simulink layout for simulation using the ID model
This model can then be used to simulate the response of the quarter-car for any desired
drivefile or shock model. A random drivefile was created using the relative amplitude
spectrum shown in Figure 4.13, with an amplitude of 1 inch on the wheelloader and an
amplitude of 100 lb and a mean of 500 lb on the aeroloader. This drivefile was run for all
12 different shock configurations described in Chapter 3 using both the original model
and the ID model. To compare the responses from the two models, the standard
deviation was calculated for each signal. A comparison of the tire force standard
deviation for the simulated data and the ID model for all 12 shock configurations is
shown in Figure 4.14. As this figure shows, the tire force standard deviation trend is
accurately predicted by the ID model. The error in tire force standard deviation is
approximately 4 lb, which is less than 1% error. Similar trends exist for the other signals.
This shows that the ID model can be used to accurately predict the response measures
and aid in shock selection.
92
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency, Hz
Rel
ativ
e A
mpl
itude
Spe
ctru
m
WheelloaderAeroloader
Figure 4.13. Relative amplitude spectra for drivefile
1 2 3 4 5 6 7 8 9 10 11 12410
420
430
440
450
460
470
480
490
500
510
Shock Configuration
Tire
For
ce S
TD, l
bs
Simulated DataID Model
1 2 3 4 5 6 7 8 9 10 11 12410
420
430
440
450
460
470
480
490
500
510
Shock Configuration
Tire
For
ce S
TD, l
bs
Simulated DataID Model
Figure 4.14. Comparison of tire force standard deviation
4.5 System Identification for Quarter-Car Rig Data
The ID methods were developed in Section 4.4 using ideal simulated data sets from a
known linear system with no sensor noise. To further prepare the method for real 8-post
rig data, the method was applied to quarter-car rig data. A series of tests were performed
93
using an existing quarter-car rig, 6 different builds of the Penske 7300 shock absorber,
and instrumentation to measure the quarter-car rig response. The data collected from the
tests was then used to identify a quarter-car model using the methods developed in
Section 4.4. To confirm that the model can be used to accurately predict the response of
the quarter-car rig, the model was used to predict the response for different shock builds,
which was then compared to test data.
4.5.1 Experimental Setup
The quarter-car rig tests were performed on an existing quarter-car rig, shown in Figure
4.14, available at the Center for Vehicle Systems and Safety (CVeSS) of Virginia Tech.
The quarter-car rig consists of a fixed external frame and two floating frames, which slide
up and down inside the fixed frame using roller bearings and Teflon bearings. Both the
fixed frame and the floating frames are constructed from 80/20® extruded aluminum.
The upper and lower floating frames are meant to represent the sprung and unsprung
masses. At the bottom of the rig, there is a hydraulic actuator, which can be used to
simulate a road excitation. The hydraulic actuator attaches to the unsprung mass through
a series of rubber bushings, selected to represent the stiffness of a passenger car tire.
The sprung mass is supported by the unsprung mass using two air springs. To provide a
larger compressible air volume, which results in a more linear air spring behavior, the air
springs were attached to a 30 gallon air tank. For our study, we also mounted a Penske
7300 shock absorber between the air springs, as shown in Figure 4.15. Before each test,
the air springs were used to set the suspension sag so that the shock had 4.5 inches of
shaft exposed. There was space to add eight 50-lb weights to the sprung mass, bringing
the total sprung mass up to about 575 lb. While this is lower than the estimated quarter-
car sprung mass of 675 lb for a NASCAR cup car, the rig can still be used to prove that
the system identification methods will work on experimental data.
94
SprungMass
UnsprungMass
AirSprings
Shock
HydraulicActuator
Rubber Bushings
SprungMass
UnsprungMass
AirSprings
Shock
HydraulicActuator
Rubber Bushings
Figure 4.15. Quarter-car rig at CVeSS (photo by author, 2009)
Sensors were mounted on the quarter-car rig to measure the rig’s response during each
test. A shock potentiometer was mounted across the shock directly to the shock bolts, as
shown in Figure 4.16b. An accelerometer was mounted on the sprung and unsprung
masses, as shown in Figure 4.16a. A string potentiometer with an integrated velocity
sensor was mounted between the sprung mass and the fixed frame, as shown in Figure
4.16c. Another string potentiometer was mounted between the unsprung mass and the
floor. The hydraulic actuator also had an integrated LVDT and a delta-P cell.
95
(a) (b) (c)
Figure 4.16. Sensor mounting: (a) Accelerometer, (b) Shock potentiometer,
(c) String potentiometer with integrated tachometer (photos by author, 2009)
All data acquisition, signal monitoring, and signal generation was done using a dSPACE
AutoBox connected to a PC running Controldesk software. The AutoBox is a real-time
system, which has 40 A/D channels and 16 D/A channels. It can also be connected to a
PC using a fiberoptic cable to transfer information between the Controldesk software user
interface on the PC and the real-time AutoBox.
All sensor signals were routed to the A/D channels on the AutoBox for data acquisition
shown in Figure 4.17. Controldesk was used to define different drivefiles to be used as a
position command for the hydraulic actuator. The signal defined in Controldesk was then
converted to an analog signal through the dSPACE D/A, and the analog position
command was sent to the MTS 407 controller, which controls the hydraulic actuator.
96
(a) (b) (c)
Figure 4.17. Other components: (a) dSPACE AutoBox,
(b) dSPACE Controldesk software, (c) MTS 407 controller (photos by author, 2009)
A series of drivefiles were run for each shock tested. Before each series of tests, a
warmup sine test was run at 0.6 inch amplitude and 2.8 Hz to heat up the shock. The first
test that was run was a 120 second random signal created by the multisine method
described in Chapter 3, with the relative amplitude spectrum shown in Table 4.3. The
spectrum was adjusted to be large enough to excite all the desired frequencies, while not
causing excessive vibrations in the rig.
Table 4.3. Relative amplitude spectrum for random drivefile
Frequency RelativeHz Amplitude0.1 11 12 0.83 0.75 0.58 0.310 0.315 0.320 0.1
To explore the steady-state response at specific frequencies, six sine tests were run at the
amplitudes and frequencies shown in Table 4.4. These frequencies were chosen to
include frequencies near, above, and below the sprung and unsprung resonance
frequencies. During sine tests, the rig was allowed to reach steady-state, then a 20 second
data block was recorded.
97
Table 4.4. Sine wave tests
Frequency AmplitudeHz in1 1
2.8 0.63.5 0.55 0.49 0.412 0.3
A bump test was also conducted by collecting a 60 second data block a using a 1 inch
square wave with a period of 20 seconds.
Six different shock configurations were tested, as shown in Table 4.5. These include all
three shock builds tested in Chapter 3 at both 1 click and 14 clicks on the bleed adjuster
and 50 psi gas pressure.
Table 4.5. Shock configurations tested
Bleed Comp HV Gas1 Build 1 soft soft hard 50 psi2 Build 1 hard soft hard 50 psi3 Build 2 soft soft soft 50 psi4 Build 2 hard soft soft 50 psi5 Build 4 soft hard 0 50 psi6 Build 4 hard hard 0 50 psi
After the data was collected, it was then processed to prepare it for the ID process. The
main processing steps were to estimate the shock velocity and shock force for each test.
To make sure the shock velocity was calculated correctly, it was calculated using three
different methods
1. Filter the shock potentiometer and differentiate,
2. Calculate the shock displacement using the string potentiometers on the sprung
and unsprung masses, differentiate, and filter, and
3. Calculate the shock velocity using the string potentiometer velocity and filter.
Filtering was done by applying an 8th order, zero-phase, Butterworth lowpass filter using
MATLAB’s filtfilt command, which filters the signal forwards and backwards to cancel
98
phase lag. Numerical differentiation was performed using MATLAB’s gradient
command, which uses central differences to reduce phase shift. These three methods
provided very consistent estimates of the shock velocity. Since the velocity calculated
from the shock potentiometer measurement will be available on the 8-post rig, we will
use it for further analysis. The shock velocity was then used as an input to the
corresponding shock model for the test, providing an estimate of the shock force.
4.5.2 Experimental ID
The ID method developed in Section 4.3 requires data from a test with a linear shock and
a test with a different shock. Since shock configuration 1 exhibits a near-linear behavior,
the data from the random test with shock configuration test was used as the linear data
set. The linear damping for shock configuration 1 was estimated to be 31.03 lb/(in/s)
using linear regression on the shock dynamometer force-velocity plot. The data from the
random test with shock configuration 1 was then used to identify the frequency response
between the actuator position and the sensor output with shock configuration 1 installed.
Once this transfer function was determined, the data from the random test for shock
configuration 5 was used to determine the transfer function between the change in shock
force and the sensor output.
The FRF estimates were calculated using 50 data blocks with a Hamming window using
50% overlap from the 120 second random signal. This corresponds to a spectral
resolution of 0.22 Hz. The parametric transfer function estimate was calculated using
invfreqs on each input-output pair. The numerator and denominator polynomials were
chosen to be 4th order, and the coefficients were fitted using the FRF estimate between
0.5 and 15 Hz, corresponding to the frequency range where the FRF estimates had good
coherence. The resulting estimates for the sprung and unsprung mass positions are
shown in Figure 4.18. The coherence for the FRF estimates is shown in Figure 4.19. The
characteristic equation for the transfer function estimate between wheelloader position
and chassis position indicated that the first mode was at 3.9 Hz with 54% damping, and
the second mode was at 8.6 Hz with 32% damping.
99
0 5 10 15 20-30
-25
-20
-15
-10
-5
0
5
Frequency, HzCha
ssis
Pos
ition
/ W
heel
load
er P
ositi
on, d
B
FRF EstimateTF Fit
0 5 10 15 20
-100
-80
-60
-40
Cha
ssis
Pos
ition
/ S
hock
For
ce, d
B
Frequency, Hz
0 5 10 15 20-15
-10
-5
0
5
Whe
el P
ositi
on /
Whe
ello
ader
Pos
ition
, dB
Frequency, Hz0 5 10 15 20
-100
-90
-80
-70
-60
Whe
el P
ositi
on /
Sho
ck F
orce
, dB
Frequency, Hz
Figure 4.18. Quarter-car rig FRF estimates
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Frequency, Hz
Coh
eren
ce
Chassis Position / Road InputChassis Position / Shock Input
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Frequency, Hz
Coh
eren
ce
Wheel Position / Road InputWheel Position / Shock Input
Figure 4.19. Coherence for FRF estimates
4.5.3 Validation
Now that we have used quarter-car rig data to create a model of the rig, we can now use
the model to predict the response of the quarter-car rig. To validate the model’s ability to
predict rig response, the predictions are compared against test data. The model was used
to run simulations using the same Simulink layout shown in Figure 4.12 using the same
drivefiles used on the quarter-car rig.
100
First, we compare the time signals directly. Since shock configurations 1 and 5 were
used to generate the model, we consider shock configuration 3 for validation purposes.
Figure 4.20 compares simulation versus test data for a 5 second time signal of the chassis
position for the random test with shock configuration 3. As this figure shows, the
simulation shows good agreement with the experimental data. Similar agreement exists
with the other sensors.
Figure 4.20. Comparison of simulated and measured time signals
Next, we compare how well our model is able to predict the change in different response
measures as the shocks are changed. Figure 4.21 compares measured and predicted
response measures for four different responses. While there is some error in predictions,
the simulation is generally effective at predicting the trend. Since the trend is what is
most commonly used in rig testing, it seems reasonable to use this model to do
preliminary shock studies to make rig testing more efficient.
101
1 2 3 4 5 60.21
0.22
0.23
0.24
0.25
0.26
Shock Configuration
Cha
ssis
Pos
ition
STD
, in
DataSimulation
1 2 3 4 5 60.26
0.265
0.27
0.275
0.28
0.285
Shock Configuration
Whe
el P
ositi
on S
TD, i
n
1 2 3 4 5 60.26
0.265
0.27
0.275
0.28
0.285
Shock Configuration
Rid
e H
eigh
t STD
, in
1 2 3 4 5 64.5
5
5.5
6
6.5
7
7.5
Shock Configuration
Sho
ck V
eloc
ity S
TD, i
n/s
Figure 4.21. Comparison of simulated versus measured response measures
4.6 Issues with Extending to Full-Vehicle Modeling
While the quarter-car rig results presented here are promising, extending this method
from this ideal environment to work for 8-post rig test data presents further challenges
that should be considered. First, there are many more potential sources of nonlinearity on
the 8-post rig. These include tire separation, coil bind, coil separation, progressive spring
rates, bump stops, nonlinear motion ratios, tire and joint friction, and play in the
suspension. Also a concern is that there are more inputs to consider, including inputs that
will be filtered across the chassis and four shocks whose force is indirectly excited from
the rig actuators.
Despite these potential concerns, there are reasons to suggest that we may overcome
these issues and be successful modeling the 8-post rig. First, the RMS-based response
measures are gross dynamic measures of the response, so even our model does not match
102
the ideal response exactly, it is still possible that our RMS trends will match. Linear
modeling has already been used to model the response of an 8-post rig test with positive
results during drivefile iteration. The weight of the vehicle and aeroloader downforce
will act to preload the tires, reducing the amount of time that the tires will separate from
the wheelloaders. Suspension joint friction and slop will often be minimal to provide
good performance.
One area where we will improve as we move to 8-post rig scale is the shock force
estimates. In this chapter, significant error may be caused by the estimation of the shock
force used in the ID process. Any error in the shock model or the velocity estimate
would cause an error in the shock force estimate, which would create an error in the
model. For the 8-post study we will directly measure shock force using a shock load cell.
Another area of improvement is the requirement that the baseline shocks are linear. We
will develop the identification method such that it will work well for both linear and
nonlinear shocks.
103
Chapter 5 Full-Vehicle Modeling and Identification
Chapters 5 and 6 extend the quarter-car modeling and identification work presented in
Chapter 4 to full-vehicle scale. This chapter focuses on performing system identification
using data generated from a known simulation model. Since we know the ideal model,
we will be able to verify that our identification methods accurately match the ideal model
behavior. Once we have shown that our identification methods can accurately match the
ideal model, we will then apply our methods to data collected from an 8-post rig
experiment in Chapter 6.
5.1 Full-Vehicle Model
This section presents the full-vehicle model which will be used as the ideal model in this
chapter. First, the linear full-vehicle model is derived. Important model dynamics are
then characterized by showing select frequency response plots. Next, we incorporate the
nonlinear shock models developed in Chapter 3 into our model. Once our model has
been fully assembled, we will then perform simulations using the model. The resulting
data sets will be used to perform system identification to generate a model that will be
compared with the original model.
5.1.1 Model Derivation
The vehicle may be modeled as 5 rigid bodies, representing the chassis and the 4 wheels,
which are connected to the chassis through the suspension members. Suspensions are
typically designed to allow for only vertical wheel motion relative to the chassis, so the
wheel rigid bodies can be simplified to a vertical mass. As discussed before, the 8-post
rig applies forces primarily in a vertical direction and constrains vehicle motion in the
yaw plane by constraining the tire contact patch to remain within the wheelloader
104
constraints. As a result of this, the yaw plane dynamics (lateral, longitudinal, yaw) are
not excited significantly in comparison with the vertical dynamics (heave, pitch, roll).
This suggests treating the chassis as having 3 inertial terms corresponding to the heave,
roll, and pitch degrees of freedom. The above reasoning motivates the selection of a 7
degree-of-freedom (7 DOF) model, with degrees of freedom for heave, pitch, and roll of
the chassis and vertical motion of the four tires.
The derivation of the 7 DOF model begins by considering the effect of a suspension force
on the sprung mass and unsprung mass. Once the general formulation is complete, this
will then be applied to model the effect of different suspension components, including
primary suspension springs, tires, shocks, and rollbars. Upon completing the formulation
for all the desired components, the forces due to each component will be combined using
superposition to form the final equations of motion.
We begin with the diagram of the chassis shown in Figure 5.1. The coordinate system is
chosen about the center of gravity, which lies a distance a behind the front axle. We
assume the center of gravity to lie along the vehicle centerline and the front and rear track
widths to be equal. The influence of external forces applied to the 3 DOF chassis model
by the aeroloaders may be represented as a heave force, roll moment, and pitch moment.
The compressive force due to the suspension at each corner is also shown.
105
y
x
z
Fz
MΦΦ
θ
Mθ
z1LF
FLF
z1RR
FRR
z1LR
FLR
z1RF
FRF
a
b
t/2
M, Jθ, JΦ
CG
y
x
z
Fz
MΦΦ
θ
Mθ
z1LF
FLF
z1RR
FRR
z1LR
FLR
z1RF
FRF
a
b
t/2
M, Jθ, JΦ
CG
Figure 5.1. Chassis geometry and forces
The geometric transformation between the chassis heave, pitch, and roll and the
displacement of the four corners of the chassis is
21
21
21
21
1
1111
LF t
RF t
LR t
RR t
azz
azbzbz
z T
θφ
− − − = −
= Φ (5.1)
Summation of forces and moments on the chassis yields
2 2 2 2
0 0 1 1 1 10 00 0
LFz
RFt t t t
LR
RR
Tc c ext
FM z F
FJ M
FJ a a b b M
F
M T F F
θ θ
φ φ
θφ
= − − + − −
Φ = +
(5.2)
106
For the unsprung masses, we define positive displacement to be upwards. This yields the
equation
2
2
2
2
2
0 0 00 0 00 0 00 0 0
LF LFLFt
RF RFRFt
LR LRLRt
RR RRRRt
t c
Fm zFm zFm zFm z
M z F
= −
= − (5.3)
Equations (5.2) and (5.3) define the effect of a generic compressive suspension force on
the sprung and unsprung masses. Next, we will consider the equations for linear springs,
tires, shocks, and anti-rollbars.
For a linear primary suspension spring, the suspension force is proportional to the
suspension displacement. This can be represented for all four corners as
( )
2 1
2 1
2 1
2 1
2 1
0 0 00 0 00 0 00 0 0
LF LFLFLF
RF RFRFRF
LR LRLRLR
RR RRRRRR
c s
F z zkF z zkF z zkF z zk
F K z z
= −
= −
(5.4)
A linear tire applies a force to the unsprung mass proportional to the tire deflection,
expressed by
2
2
2
2
0 0 00 0 00 0 00 0 0
LF LF LF LFt t RRF RF RF RF
t t RLR LR LR LR
t t RRR RR RR RR
t t R
t t R
F k z zF k z zF k z zF k z z
F K z
= −
= −( )2 z (5.5)
107
Combining Equations (5.1), (5.2), and (5.4) for the sprung mass yields
1 2
2
2
+
+
+
T Tc s s ext
T Ts s ext
T Tc s s ext
M T K z T K z F
T K T T K z F
M T K T T K Fz
Φ = − +
= − Φ +
Φ Φ = −
(5.6)
Combining Equations (5.1), (5.3), (5.4), and (5.5) for the unsprung mass gives
( ) ( )( )( )
( )
2
2 1 2
1 2
2
22
- t c t
s t R
s s t t R
s s t t R
t s s t t R
M z F FK z z K z z
K z K K z K z
K T K K z K z
M z K T K K K zz
= − +
= − + −
= − + +
= Φ − + +
Φ = − + +
(5.7)
Equations (5.6) and (5.7) can be combined to provide
22
0 00 0
0
0
T Tc exts s
t t Rs s t
extP
t R
M I FT K T T KM Kz zz K T K K
I FMZ K Z
K z
Φ Φ − + = − +
+ =
(5.8)
The equations for linear shocks in parallel with the suspension and tire springs can easily
be derived using similarity to the above equations. Assuming tire damping to be
negligible, the damping matrix is
T Ts s
Ps s
T C T T CC
C T C −
= − (5.9)
where sC is a diagonal matrix containing the damping coefficients for the four corners.
The resulting equations of motion including linear springs, shocks, and tire stiffnesses are
108
00
extP P
t R
I FMZ C Z K Z
K z
+ + =
(5.10)
The next component that we wish to include is the rollbar. The purpose of an anti-rollbar
is to increase the roll rate of an axle without increasing the vertical ride rate. The anti-
rollbar on an axle does this by applying a restoring roll moment proportional to the
chassis roll relative to the axle roll. The forces applied by the anti-rollbar for a positive
roll angle are shown in Figure 5.2.
z1Lz1
R
z2Lz2
Rt/2 t/2
θ
θAXLE
F F
F F
z1Lz1
R
z2Lz2
Rt/2 t/2
θ
θAXLE
F F
F F
Figure 5.2. Forces applied due to anti-rollbar
The relative roll angle is defined as
2 2
1 12
2
1
L R
rel axle
Lrel t t
R
z zt
zz
θ θ θ θ
θθ
−= − = −
= −
(5.11)
The force applied to the suspension by the anti-rollbar can be found by
( )1
22 2
2
R relt
LR R R
R
F k
k k k zt t t
z
θ
θ
=
= −
(5.12)
It can be seen that the anti-rollbar applies a compressive force to the right side of the
chassis and a tension force to the left side of the chassis for a positive relative roll angle.
109
We can then write the suspension forces due to the anti-rollbar in terms of the vehicle
position vector as
2 2
2 2
22 2
2 2
1
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
F F FR R R
F F FLF R R R
RFR R R
LR R R R
RRR R RR R R
c
k k kt t t
F k k kF t t tF zk k kF t t t
k k kt t t
F K Z
− −
− Φ = − −
−
= (5.13)
where FRk and R
Rk are the front and rear anti-rollbar stiffnesses. Using Equations (5.2)
and (5.3), the contribution to the stiffness matrix due to the anti-rollbar is
1
1
- T
rollT K
KK
=
(5.14)
This stiffness matrix can be added to the existing stiffness matrix in Equation (5.10) to
include the effects of rollbars. This gives us the new equations of motion
( )0
0ext
P rollt R
I FMZ CZ K K Z
K z
+ + + =
(5.15)
The last component that we wish to include is an arbitrary suspension force. This will be
used to include nonlinear shock models in our simulations, but it could also be used for
any other arbitrary suspension force. Defining our external suspension force to be FD, we
can find the effect of an arbitrary suspension force on the vehicle response using
Equations (5.2) and (5.3) as
2
0
0
Tc
Dt
M TF
M z I Φ
= − (5.16)
110
Combining Equations (5.15) and (5.16) yields the final equation of motion
( ) 0
0
extT
P P roll Rt
D
P
FI T
MZ C Z K K Z zK I
F
MZ C Z KZ Lu
+ + + = −
+ + =
(5.17)
Where the stiffness and damping matrices can be found using
1
1
-
T T Ts s
s s t
T Ts s
Ps s
T K T T K T KK
K T K K K
T C T T CC
C T C
−= + − + −
= −
(5.18)
The matrices Ks, Kt, and Cs are 4x4 diagonal matrices containing the spring stiffness, tire
stiffness, and linear damping for the four corners. The matrix K1 is defined in Equation
(5.13), and the transformation matrix T is defined in Equation (5.1). The mass matrix is a
7x7 diagonal matrix containing the inertia of the 7 degrees of freedom.
The equations of motion can be rewritten in state space form as
1
2
1 11 1 1
2 2
0 0
P
x Z
x Z
x I xu
x M K M C x M L
x Ax Bu
− − −
=
=
= + − −
= +
(5.19)
The matrices A and B can be calculated using Equations (5.1), (5.13), and (5.17)-(5.19).
In addition to defining the dynamic equations, we must also define the output equation
y Cx Du= + (5.20)
111
These outputs represent the signals recorded during an 8-post rig test. The most
important model outputs are the shock velocities, as the shock velocities will be used as
inputs to the shock models, which will be fed back as an input to the linear vehicle
model. The shock velocity may be calculated as
[ ]
2 1
2
4 7 40
shock
x
v z zz T
T I x
= −
= − Φ
= −
(5.21)
The next set of outputs that we will consider is the tire forces. This is an important
output for an 8-post rig test simulation, as tire force is often used to calculate a “grip
number” metric, which is used as an indication of the quality of mechanical grip. The tire
force may be calculated as
( )[ ] [ ]
2
4 3 4 7 4 3 4 40 0 0 0t t R
x t x x t x
F K z z
K x K u
= − +
= + (5.22)
The last output equation we show is for ride height. Ride height is a critical measurement
during a rig test, as it is used to indicate how much the aerodynamic forces will vary. We
define front and rear ride heights as the displacement of the nose and spoiler along the
vehicle centerline relative to the ground. To define these ride heights, it is first necessary
to approximate the height of the ground along the vehicle centerline. First, the ground
height at the center of the front and rear axles may be approximated to be the averages of
the front and rear wheelloader positions, respectively. Using these two ground heights
and assuming the ground height varies linearly along the centerline, the ground height
below the nose and wing may be determined. Front and rear ride height may then be
calculated as
( )( )
( ) ( )( ) ( )
1 1
2 2 2 2
12 11
2
2 2 2 2
2 3 2 4
2 2 2 2
1 0 0
1 0
0 0l l
x
l l l l l ll l l l
x xl l l l l ll l l l
a lFRHx
b lRRH
u+ +
+ +
− + = +
− − + − −
(5.23)
112
Where l is the wheelbase, l1 is the distance from the front axle to the nose, and l2 is the
distance from the rear axle to the spoiler.
In addition to the output equations shown above, additional output equations were
constructed for the state vector, suspension deflections, chassis accelerations, wheel
accelerations, and spring forces.
The model parameters used are shown in Table 5.1. The sprung and unsprung masses
correspond to a 3000 lb car with 85% sprung mass, and 45% of the unsprung mass in the
front. The roll and pitch radii of gyration are 40% and 60% of the track and wheelbase,
respectively. The suspension and tire stiffnesses are within the range of typical values.
While these parameters do not precisely reflect those of a NASCAR Cup car, they do
describe a vehicle whose behavior is qualitatively similar, making it sufficient for ID
method development.
Table 5.1. Model parameters
Track Width 70 inCG-Front Axle 60 inCG-Rear Axle 50 inSprung Mass 2550 lbmRoll Inertia 1.5E+06 lbm-in^2Pitch Inertia 5.5E+06 lbm-in^2Front Unsprung Mass 101 lbmRear Unsprung Mass 124 lbmFront Spring Rate 1280 lbf/inRear Spring Rate 588 lbf/inTire Stiffness 2000 lbf/inLinear Damper 30 lbf/(in/s)
5.1.2 Frequency Response
This section discusses the frequency response for the linear vehicle model presented in
the previous section. These plots provide insight on system behavior, which can be
applied during the identification process. The outputs considered include shock velocity,
tire force, and ride height. We only consider 3 of the 11 potential inputs, but the
observations below may be generalized to the remaining 8 inputs. The inputs considered
113
include chassis heave force, left front wheelloader position, and left front external shock
force.
The first set of outputs that we consider are the shock velocities, as shown in Figure 5.3.
For the chassis force input, 1000 lb chassis heave force gives 10 in/s near resonance, and
1 in/s below 1 Hz and above 9 Hz. Further chassis velocity can be attained by aeroloader
force by applying roll and pitch moments.
The left front wheelloader input creates about 1 in/s shock velocity at low frequency and
30 in/s shock velocity near resonance for a 1 inch left front wheelloader input. The left
front shock velocity frequency response maintains a high level at higher frequencies due
to the proximity of the wheelloader to the shock. All other shock velocities drop about
20 dB at higher frequencies due to the filtering effect of the chassis.
The left front external shock force produces about 1 in/s left front shock velocity for
every 100 lb. The other shock velocities have similar levels at lower frequencies, but the
level drops 40 dB at higher frequencies due to the filtering effects of the chassis.
5 10 15 20-90
-80
-70
-60
-50
-40
-30
Frequency, Hz
Sho
ck V
eloc
ity /
Hea
ve F
orce
, dB
5 10 15 20-30
-20
-10
0
10
20
30
40
Frequency, Hz
Sho
ck V
eloc
ity /
LF W
heel
load
er P
ositi
on, d
B
5 10 15 20-120
-100
-80
-60
-40
-20
Frequency, Hz
Sho
ck V
eloc
ity /
LF S
hock
For
ce, d
B
LF Shock VelocityRF Shock VelocityLR Shock VelocityRR Shock Velocity
Figure 5.3. Shock velocity frequency response
The next set of outputs that we consider is the tire forces. For the chassis heave force
input, 1000 lb of chassis heave force causes about 500 lb tire force near resonance, 300 lb
of force at low frequency, and 100 lb at 10 Hz.
For the left front wheelloader position input, a 1 inch input produces 100 lb of tire force
at low frequency, and as much as 3000 lb near resonance. As frequency increases from
resonance, the left front tire force maintains a high level, while the other tire forces drop
due to the filtering effect of the chassis.
114
For the left front external shock force input, a 100-lb shock force input creates 10 lb tire
force at low frequency, and as much as 100 lb at resonance. The left front tire force stays
at a high level at higher frequencies, while the other tire forces drop 40 dB.
5 10 15 20-35
-30
-25
-20
-15
-10
-5
0
Frequency, Hz
Tire
For
ce /
Hea
ve F
orce
, dB
5 10 15 2030
40
50
60
70
Frequency, Hz
Tire
For
ce /
LF W
heel
load
er P
ositi
on, d
B5 10 15 20
-60
-50
-40
-30
-20
-10
0
10
Frequency, Hz
Tire
For
ce /
LF S
hock
For
ce, d
BLF Tire ForceRF Tire ForceLR Tire ForceRR Tire Force
Figure 5.4. Tire force frequency response
The last set of outputs we consider is the front and rear ride heights. For the chassis
heave force, 1000 lb of heave force provides 0.3-1 inches of ride height variation from 0-
5 Hz. Above 5 Hz, the response rolls off at about 40 dB/decade.
For the left front wheelloader input, the response is small at lower frequencies as the
chassis moves in phase with the wheelloaders. At 5 Hz, the response peaks, giving 1 inch
front ride height variation and 0.3 inches of rear ride height variation for a 1 inch input.
At higher frequencies, the front ride height maintains a high response level, while the rear
ride height drops to -40 dB due to the filtering effect of the chassis.
5 10 15 20-100
-90
-80
-70
-60
Frequency, Hz
Rid
e H
eigh
t / H
eave
For
ce, d
B
5 10 15 20-50
-40
-30
-20
-10
0
10
Frequency, Hz
Rid
e H
eigh
t / L
F W
heel
load
er P
ositi
on, d
B
5 10 15 20-140
-120
-100
-80
-60
Frequency, Hz
Rid
e H
eigh
t / L
F S
hock
For
ce, d
B
Front RHRear RH
Figure 5.5. Ride height frequency response
These observations can be used to assist our identification efforts for these key outputs.
To successfully perform nonparametric frequency response estimation, we need to have
significant output energy at all frequencies of interest relative to the noise level. This will
115
provide a good signal-to-noise ratio, or coherence near 1 for a linear system. To design
excitation to provide good output signal levels, the system behavior should be considered.
Notice that the chassis heave force is effective for exciting response below 5 Hz, but the
response drops off at higher frequencies. This trend is also true for chassis roll and pitch
moments. Conversely, the wheelloaders are often ineffective at exciting outputs below 3
Hz but are useful for higher frequency excitation. Also, the most effective excitation at
higher frequency will be the wheelloader closest to the sensor of interest. This suggests
the need for excitation from both the closest wheelloader and the aeroloaders if we want a
good signal-to-noise ratio from 1 to 20 Hz.
While it is ideal to get high signal-to-noise ratios in all output signals at all frequencies,
this is often impractical due to the system’s behavior, physical input limitations, and
vehicle response limitations. If the output level of a particular sensor due to a particular
input will always be low over a range of frequencies for the desired excitation signals to
be used with the model, the poor frequency response estimation for that input-output pair
in that frequency range may be replaced by a small or zero frequency response with
minimal reduction in model accuracy from the ideal model.
Notice that accurate frequency response estimation for aeroloader inputs is only critical at
lower frequencies. At higher frequencies, the response is attenuated, so a poor frequency
response estimate could be replaced by a small or zero frequency response to reduce
modeling error.
For the wheelloader inputs and shock force inputs, the frequency response is highest near
resonance and remains at a high level at higher frequencies for input-output that are
directly connected within one corner and not connected through the chassis. This
indicates that frequency response estimation is most important for wheelloader and shock
force inputs near resonance. It is also important for directly-coupled input-output pairs at
higher frequencies. Modeling error due to low response levels outside these regions can
be replaced with a small or zero frequency response.
116
This section has summarized the full-vehicle model’s behavior for the key outputs of
shock velocity, tire force, and ride height. These results can then be applied to design
excitation signals for identification tests and to modify the identification process where
frequency response estimation is poor.
5.1.3 Full-Vehicle Simulation
Now that we have developed a full-vehicle model and explored the behavior of the
model, we will now apply the model to generate data sets that can be used to develop the
identification method. Since we already know the ideal model, performing identification
on data generated by the model will allow us to validate our identified model with the
ideal model, which will allow us to correct any issues in the identification process.
First, we need to incorporate the shock absorber models developed in Chapter 3 into the
vehicle model. This is accomplished by applying the shock velocity calculated from the
linear vehicle model to the nonlinear dynamic shock model, as shown in Figure 5.6. The
shock force calculated by the shock model is then fed back as an input to the linear
vehicle model using the external shock force input. This method is identical to the
method in Chapter 4, except there are 4 shock models instead of just one.
LF Shock Model (θLF)
Linear Vehicle Model
RF Shock Model (θRF)
LR Shock Model (θLR)
RR Shock Model (θRR)
LF Shock Velocity
RF Shock Velocity
LR Shock Velocity
RR Shock Velocity
Outputs
LF Shock Force
RF Shock Force
LR Shock Force
RR Shock Force
Aeroloader Forces
Wheelloader Positions
Shock Forces
Complete Model
LF Shock Model (θLF)LF Shock Model (θLF)
Linear Vehicle Model
RF Shock Model (θRF)RF Shock Model (θRF)
LR Shock Model (θLR)LR Shock Model (θLR)
RR Shock Model (θRR)RR Shock Model (θRR)
LF Shock VelocityLF Shock Velocity
RF Shock VelocityRF Shock Velocity
LR Shock VelocityLR Shock Velocity
RR Shock Velocity
Outputs
LF Shock Force
RF Shock Force
LR Shock Force
RR Shock Force
Aeroloader Forces
Wheelloader Positions
Shock Forces
Complete Model
Figure 5.6. Coupling of vehicle and shock absorber models
This model structure can be easily implemented using Simulink. The equations for the
LPNL2 nonlinear dynamic shock absorber are described in Chapter 3. To implement the
Simulink model in discrete-time, the linear vehicle model and the shock model transfer
117
functions are discretized at 100 Hz using Tustin’s bilinear transformation, available in
MATLAB’s c2d function. To make Simulink model more organized, the shock models
are arranged in subsystems, as shown in Figure 5.7.
Ysim
To Workspace
y _linear
F_poly
F_LPF1
F_LPNL2
Shock Models
U
Rig Drivefiley(n)=Cx(n)+Du(n)
x(n+1)=Ax(n)+Bu(n)
Linear Fullcar
u_linear
u_linear
F_poly
F_LPF1
F_LPNL2
y _linear
y _linear
3F_LPNL2
2F_LPF1
1F_poly
U U(E)
RR shock velocity
v
F_poly linear
F_LPF1F_LPNL2
RR Shock Model
U U(E)
RF shock velocity
v
F_poly linear
F_LPF1F_LPNL2
RF Shock Model
U U(E)
LR shock velocity
v
F_poly linear
F_LPF1F_LPNL2
LR Shock Model
U U(E)
LF shock velocity
vF_poly linear
F_LPF1
F_LPNL2
LF Shock Model
1y_l inear
v
3F_LPNL2
2F_LPF1
1F_polylinear
MATLABFunction
Polylinear
MATLABFunction
LPNL2 Weighting
bb4(z)
aa4(z)First-Order Lag
1v
F_lagF_LPNL2
F_poly
Simulink Model
Shock Model Subsystem
RR Shock Model Subsystem
Ysim
To Workspace
y _linear
F_poly
F_LPF1
F_LPNL2
Shock Models
U
Rig Drivefiley(n)=Cx(n)+Du(n)
x(n+1)=Ax(n)+Bu(n)
Linear Fullcar
u_linear
u_linear
F_poly
F_LPF1
F_LPNL2
y _linear
y _linear
3F_LPNL2
2F_LPF1
1F_poly
U U(E)
RR shock velocity
v
F_poly linear
F_LPF1F_LPNL2
RR Shock Model
U U(E)
RF shock velocity
v
F_poly linear
F_LPF1F_LPNL2
RF Shock Model
U U(E)
LR shock velocity
v
F_poly linear
F_LPF1F_LPNL2
LR Shock Model
U U(E)
LF shock velocity
vF_poly linear
F_LPF1
F_LPNL2
LF Shock Model
1y_l inear
v
3F_LPNL2
2F_LPF1
1F_polylinear
MATLABFunction
Polylinear
MATLABFunction
LPNL2 Weighting
bb4(z)
aa4(z)First-Order Lag
1v
F_lagF_LPNL2
F_poly
Simulink Model
Shock Model Subsystem
RR Shock Model Subsystem
Figure 5.7. Full-vehicle model implemented in Simulink
Five simulations were performed to generate data for use in system identification. An
800 second Gaussian bandlimited white noise drivefile was constructed to excite the
Simulink model. Each signal was bandlimited from 0 to 30 Hz using an 8th order
discrete-time Butterworth filter. The peak signal amplitudes were set to 1 inch for the
wheelloaders and 500 lb for the chassis heave force. The chassis roll moment and chassis
pitch moment amplitudes were set to be 3 times the chassis roll and pitch stiffness
respectively in in-lb/degree. This drivefile was applied to the Simulink model to generate
data that will be used to perform system identification on the linear vehicle model. One
baseline simulation was performed with a linear shock with a damping coefficient of 20
lb/(in/s) and four additional simulations were performed where the baseline shock at each
corner was replaced by a nonlinear dynamic shock model with a damping coefficient of 8
lb/(in/s).
118
5.2 System Identification Methods
This section presents system identification methods that will be applied to the data
generated using the full-vehicle model. The presentation here builds upon the work
presented for the quarter-car study in Section 4.3 and extends it for multiple shocks. To
analyze the identification methods, it will be convenient to partition the vehicle model
derived in Section 5.1 as shown in Figure 5.8. This diagram partitions the linear system
that we wish to identify into the response caused by the actuators and the additional
response caused by the external shock force, which is defined by a feedback loop, as
shown in Figure 5.8.
Hrig
y
urig
ushocks Hshocks
+
yrig
yshocks
SShock Models
vshocks
Selector
ushocks
Linear System to Identify
Hrig
y
urig
ushocks Hshocks
++
yrig
yshocks
SShock Models
vshocks
Selector
ushocks
Linear System to Identify
Figure 5.8. Diagram of system to identify
The B matrix in the state-space vehicle model in Equation (5.19) may be partitioned by
the columns corresponding to inputs due to the rig and inputs due to the external shock
force as
ext
R
D
extrig shocks D
R
rig rig shocks shocks
Fx Ax B z
F
FAx B B F
z
x Ax B u B u
= +
= + +
= + +
(5.24)
The D matrix in the state-space output equation in Equation (5.20) may be similarly
partitioned as
119
rig rig shocks shocks
y Cx DuCx D u D u
= += + +
(5.25)
The transfer functions for this partitioned system may be found by taking the Laplace
transform of Equation (5.24) and combining it with Equation (5.25), which gives us
( ) ( ) ( )( ) ( ) ( ) ( )
1 1
rig rig rig shocks shocks shocks
rig rig shocks shocks
y s C sI A B D u C sI A B D u
H s u s H s u s
− − = − + + − + = +
(5.26)
The shock velocities may be selected from the output vector using a selection matrix
shocksv Sy= (5.27)
The above notation will be used to discuss the identification methods.
5.2.1 Nonparametric Frequency Response Estimation
If all the inputs were independent, a common practice is to create a nonparametric
estimate of the frequency response matrix, which consists of a complex number for each
input, output, and frequency. For each frequency and output, we wish to fit the model
rig rig shocks shocksy H u H u= + (5.28)
where the inputs and outputs consist of the frequency domain component for a given
frequency. To fit the model, we collect several data blocks and select the model
coefficients that minimize some measure of model error. Data blocks may be taken from
a series of tests or from selecting multiple blocks of data from a longer data set. The
measured output in the kth data block will contain some error from the ideal model’s
estimate, which may be written as
[ ] [ ] [ ] [ ]rig rig shocks shocksy k H u k H u k kε= + + (5.29)
If we collect K data blocks, we can arrange the results in matrix form as
120
[ ]
[ ]
[ ] [ ]
[ ] [ ]
[ ]
[ ]
1 1 1 1
T Trig shocks T
rigT
T T shocksrig shocks
y u uH
Hy K u K u K K
Y UH E
ε
ε
= +
= + (5.30)
We can choose to select our frequency response estimate to minimize the error measure
*J E E= (5.31)
This is a least-squares optimization problem, whose solution satisfies the normal
equations
( )* *
uu uy
U U H U Y
S H S
=
= (5.32)
The frequency response estimation produced by this method is often called a H1
frequency response estimation. To find a unique solution for the frequency response
matrix H in Equation (5.32), the autospectrum matrix must be nonsingular. A necessary
condition for the autospectrum matrix to be nonsingular is that the input matrix U defined
in Equation (5.30) must be full rank, which is rank 11 in our case. This requires that
there must be at least 11 linearly independent input block rows and 11 linearly
independent input columns to solve for H.
For the simple case of linear shocks, we can show that the shock force inputs will be
linearly dependent on the rig inputs. For linear shocks, the shock force can be written
shocks shocksu Cv
CSy
=
= (5.33)
where C is a 4x4 diagonal matrix whose diagonal entries contain the 4 shock damping
coefficients. Substituting Equation (5.28) and solving for the shock force yields
121
( )( )
( ) 1
shocks
rig rig shocks shocks
shocks shocks rig rig
shocks shocks rig rig
shocks A rig
u CSy
CS H u H u
I CSH u CSH u
u I CSH CSH u
u H u
−
=
= +
− =
= −
=
(5.34)
This shows that the shock force is linearly dependent on the rig inputs when the shocks
are linear, so no unique solution to Equation (5.32) exists when this frequency response
estimation is applied in a straightforward manner. Physically, this means that since the
shock inputs are linearly related to the rig inputs, it is impossible to determine if the
outputs are linearly related to the shock inputs or the rig inputs. This argument can be
extended to shocks that exhibit linear dynamic behavior by allowing the C matrix to be
complex.
If the shock is nonlinear, the nonlinear portion of the shock force provides some shock
force excitation that is linearly independent from the rig inputs. If the nonlinear
contribution is small, the output caused by the independent portion of the shock force will
be small, causing poor coherence. This suggests one option to improve coherence is to
use highly nonlinear shock absorbers, but it may not be practical to test with a shock with
enough nonlinearity to achieve the desired model quality.
The observation that it is impossible to identify the shock input frequency response using
conventional procedures and that poor coherence is expected when the shock is close to
linear motivate developing alternate identification methods.
5.2.2 Frequency Response Estimation for Shock Force Inputs
To get good estimates of the frequency response between shock force inputs and sensor
outputs, more shock force signal energy must be linearly independent from the rig inputs.
This section provides a summary of potential methods to get quality shock force input
frequency response estimations.
122
The simplest method of applying uncorrelated shock force is to test with a highly
nonlinear shock. This will only be successful if the nonlinear portion of the shock force
can excite outputs with good signal-to-noise ratio and the nonlinear shock does not excite
vehicle nonlinearities, invalidating the linear vehicle model assumption.
The most direct method of applying linearly independent shock force is to replace the
shock with an active or semi-active suspension element and applying independent control
inputs. Alternately, different control policies could be applied for different data blocks
[38, 39]. This method has the potential for good quality frequency response estimates,
but it requires preparation of active or semi-active suspension members to be mounted in
the place of the shocks, limiting the convenience of applying this method to an arbitrary
vehicle during normal 8-post testing operations.
A convenient method motivated by the option of applying multiple control policies is to
test with multiple shocks. This method is equivalent to applying multiple control policies
using an active element, where the control policy is limited by the performance that can
be achieved by a passive shock. This will produce linearly independent input columns in
the U matrix even if the shocks are linear.
To prove that this will allow for unique identification even if the shocks are linear, we
will describe an experimental procedure that will create an input matrix that is full rank.
First let us partition the shock force into a linear shock force and an additional arbitrary
shock force
( )shocks shocksu CSy u CSy
CSy F
= + −
= + ∆ (5.35)
where C is a 4x4 diagonal matrix containing baseline linear damping coefficients that
are free to be set as desired. The arbitrary portion of the shock force may be written as
123
1 11 1
2 22 2
3 33 3
4 44 4
shocks s
shocks s
shocks s
shocks s
f u c vf u c v
Ff u c vf u c v
∆ − ∆ − ∆ = = ∆ − ∆ −
(5.36)
where ishocksu , ic , and i
sv are the shock force, baseline damping coefficient, and the shock
velocity of the ith shock. Equation (5.28) can then be rewritten
( )( )
( ) ( )1
rig rig shocks
shocks rig rig shocks
shocks rig rig shocks
rig rig shocks
y H u H CSy F
I H CS y H u H F
y I H CS H u H F
y H u H F
−
= + + ∆
− = + ∆
= − + ∆
= + ∆
(5.37)
To identify both rigH and shocksH , we need to excite both the rig inputs and the change in
shock force input. Let us collect 7 baseline data sets with linear shocks whose damping
coefficients are defined by C . Further, let the drivefiles for all 7 data sets be linearly
independent at all frequencies, such that
[ ]
[ ]
1
7
Trig
rigT
rig
uU
u
=
(5.38)
is full rank for all frequencies. Since the damping force for these 7 baseline data blocks
is CSy , the corresponding F∆ is zero.
Next, we collect 4 additional data sets, using alternate shock selections for each data
block. For the first data block, we remove the linear left front shock and replace it with a
shock with a different damping coefficient '1c , such that
124
11 1
'1 1 1
0 0, 0
0 00 0
sf c v
F c c c
∆ ∆ ∆ = = ∆ = − ≠
(5.39)
The arbitrary contribution of the shock force 1f∆ will be nonzero at all frequencies that 1sv
is nonzero. It is possible to excite the vehicle with a drivefile 1Tu such that the shock
velocity 1sv is nonzero at all frequencies except at frequencies where all inputs have a
zero FRF. Since the shock velocity frequency response consists of a ratio of polynomials
with a numerator polynomial of order 13 or less, there will be at most 13 frequencies
where the shock velocity must be zero. In practice, there will be much less frequencies
where the shock velocity must be zero, and the presence of a poor FRF estimate at a few
zeroes does not significantly degrade the model quality.
This is repeated for all four shock locations, where the shock at the present location is
replaced by an arbitrary shock and all other shocks are the baseline shocks. Collecting
the data from all 11 experiments yields the system of equations
[ ]
[ ]
[ ]
[ ]
0 0
0 0
1 11 1
2 22 2
3 33 3
4 44 4
0 0 0 01 1
0 0 0 07 70 0 0
0 0 00 0 00 0 0
rig
TT rig
TT shocks
T
T
yU
yH
u fyH
u fyu fyu fy
Y
ε
εεεεε
= +∆ ∆
∆
∆
= UH E+ (5.40)
where y0[k] is the output measured for baseline shock experiment k, and yi is the output
measured from alternate shock experiment i.
Let us restrict the following analysis to frequencies where all actuator inputs do not have
a zero for the current shock velocity. Since rigU is full rank, the first 7 columns of the
125
input matrix are rank 7. Since 0if∆ ≠ the last four columns of the input matrix are rank
4. For the input matrix to be rank deficient, at least one of the last four columns must be
linearly dependent on the first 7 columns. This implies that the equation
0rigT
i i
Ud
u f
= ∆ (5.41)
must be satisfied for some i. Since rigU is full rank, d must be zero. This requires
0if∆ = , which contradicts Equation (5.38). Therefore, an experiment conducted using
the above procedure will produce a full rank input matrix at all frequencies where the
shock velocity is nonzero, allowing estimation of a unique frequency response
estimation.
The least-squares problem defined by the data matrices in Equation (5.40) and the
objective function in Equation (5.31) may then be used to solve the normal equations in
Equation (5.32) to give the frequency response estimates for the rig inputs and the shock
inputs. In practice, more than 11 data sets would be collected to allow for averaging and
to provide an estimate of model quality.
In contrast to the above global estimation approach, it is often desirable to first fit a
baseline model for the rig inputs, then augmenting the baseline model with component
sensitivity models as needed. If the error in the baseline model is small, the results will
be very similar. An advantage of this sequential approach is that it allows us to deal with
smaller input matrices, making the solution more computationally efficient. Another
advantage is that it allows us to calculate coherence for a single-input single-output FRF
estimation for the component inputs, giving us a good indication of where the model
performs well.
The sequential approach may be written as a sequential least-squares problem by
partitioning Equation (5.40) into the rig input tests and the component input tests. The
least-squares problem for the rig inputs with K data sets may be written
126
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
*
0 0
0 0
min
1 1 1
7
rig rig
Trig
Trig
Trig
Trig rig rig rig
J E E
y uH
y K u K
y U H E
ε
ε
=
= +
= +
(5.42)
The solution to this least-squares problem satisfies
( )* *Trig rig rig rig rigU U H U Y= (5.43)
Once the rig input frequency response has been determined, the shock input frequency
responses may be estimated independently. The least-squares problem for the ith shock
input with Ki data sets, given the rig input frequency response, may be written
[ ]
[ ]
[ ] [ ]
[ ] [ ][ ]
[ ]
[ ]
[ ]
*min
1 1 1
1 1
rig
i
i
i i iH
T Ti i i rig
T Ti i i i i i rig
i iT
shock
i i i i
Ti i shock i
J E E
y y u H
y K y K u K H
fH
f K K
Y F H E
ε
ε
=
∆ − = ∆ − ∆ = + ∆
∆ = ∆ +
(5.44)
whereishockH is the frequency response for the ith shock input. This least-squares problem
satisfies
( )* *i
Ti i shock i iF F H F Y∆ ∆ = ∆ ∆ (5.45)
127
This sequential procedure was applied to the data generated by simulation, providing
frequency response estimates for both rig inputs and shock inputs.
If the baseline shocks are nonlinear, a system of equations similar to Equation (5.40) may
be derived with more nonzero entries in the last 4 columns of the input matrix. Due to
the additional nonzero entries in the input matrix, the equations cannot be decoupled as in
Equations (5.22)-(5.25), so the entire system must be solved simultaneously.
5.2.3 Parametric System Identification
Now that the nonparametric frequency responses have been estimated, these estimates
may be used to fit a parametric model. The parametric model will have the advantages of
smoothing the nonparametric frequency response, improving the model in small
frequency ranges where the nonparametric model was poor, and being more efficient for
simulation.
There are many different methods of performing parametric system identification, which
can be chosen depending on the particular application. The transfer function for each
input/output pair can be defined by its poles, zeros, and constant gain. In an ideal linear
system, all input/output pairs should share the same poles and only differ by the zeros and
constant gain. The identification of the system poles is often important for applications
in controls and structural parameter identification.
In our case, it is not necessary to identify a consistent set of system poles. All we want is
to simplify our nonparametric frequency response estimates into a convenient transfer
function relationship that will allow us to simulate the linear system more efficiently.
This allows us to use a simpler method of fitting a single-input single-output (SISO)
transfer function to the frequency response estimate for each input/output pair. This
method will give a different set of poles, zeros, and a constant for each input/output pair.
One advantage of this method is that it is very easy to fit a SISO transfer function, and
many methods are readily available.
Another advantage is that it localizes error in the frequency response estimate to the
corresponding transfer function estimate. If there is significant error in the frequency
128
response estimate for one of the input/output pairs due to factors such as noise, poor
excitation, nonlinearity, or sensor malfunction, the error will only affect the
corresponding transfer function estimate, not any other transfer function estimate.
The SISO method chosen for this work uses the invfreqs function in MATLAB [40].
This method fits a SISO transfer function of the form
( ) ( )( )
11 0
11 0
n nn nm m
m
B s b s b s bH sA s s a s a
−−
−−
+ + += =
+ + + (5.46)
by minimizing the equation-error cost function
( ) ( )( )( )( )( )
22
2t
k
B j f kJ w k h k
A j f k
π
π= −∑ (5.47)
where ( )f k is the frequency at index k, ( )h k is the nonparametric frequency response
estimate, and ( )tw k is a weighting factor. This optimization problem is nonlinear in
nature and must be solved using iterative nonlinear programming techniques. The
invfreqs function uses a Gauss-Newton iterative search to solve the nonlinear
programming problem.
5.3 System Identification Results
This section applies the identification methods described in Section 5.2 with the
simulated data sets described in Section 5.1 to generate a model that approximates the
vehicle behavior. The resulting model is then compared with the original model to
determine where the ID process was successful and to help improve the ID method.
5.3.1 Nonparametric Identification Results
Five simulations were performed to generate data for use in system identification. An
800 second Gaussian white noise drivefile, bandlimited from 0 to 30 Hz drivefile was
129
constructed to excite the Simulink model. One baseline simulation was performed with
an empirical shock model with a damping coefficient of 20 lb/(in/s) and four additional
simulations were performed where the baseline shocks were replaced by a different
empirical shock model with a damping coefficient of 8 lb/(in/s).
The first data set was used to estimate the rig input frequency response. Forty data
blocks were created from the 800 second data set using a Hamming window with 50%
overlap, giving frequency resolution of 0.03 Hz. The frequency response estimates for
the rig inputs was very close to the ideal frequency response, with coherence very close
to 1. The only exception was the ride height / roll moment frequency response, which
had almost no response due to vehicle symmetry.
The next four data sets were used to estimate the shock input frequency response, using
the same settings as the first data set. The coherence and frequency response estimate for
the left front shock velocity output for all four shock force inputs is shown in Figure 5.9.
The results for the left front shock velocity are consistent with the results for the other
three shock velocities. For the left front shock force input, the frequency response
estimate is accurate for all frequencies except near the anti-resonance at 6 Hz. For the
other shock force inputs, the frequency response estimate is only good near the resonant
peaks.
0 5 10 15 20-55
-50
-45
-40
-35
-30
-25
Frequency, Hz
LF S
uspe
nsio
n V
eloc
ity /
LF S
hock
For
ce, d
B
IdealEstimate
0 5 10 15 20-110
-100
-90
-80
-70
-60
-50
-40
-30
LF S
uspe
nsio
n V
eloc
ity /
RF
Sho
ck F
orce
, dB
Frequency, Hz0 5 10 15 20
-100
-90
-80
-70
-60
-50
-40
-30
LF S
uspe
nsio
n V
eloc
ity /
LR S
hock
For
ce, d
B
Frequency, Hz0 5 10 15 20
-90
-80
-70
-60
-50
-40
-30
LF S
uspe
nsio
n V
eloc
ity /
RR
Sho
ck F
orce
, dB
Frequency, Hz (a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz
(b)
Figure 5.9. Shock velocity FRF estimate: (a) FRF estimate, (b) Coherence
130
In all cases, the estimate is poor only where there is very low output energy, resulting in
poor coherence. This is a significant observation, because if the model is poor when the
response level is low, the poor frequency response estimate may be replaced by a small
frequency response, which will be close to the ideal model.
Next we show the coherence and frequency response estimate for the left front tire force,
which is consistent with the other three tire forces. For the left front shock force, we see
that the estimate is accurate except at lower frequencies, corresponding to the low
response level. For the other shock force inputs, the estimate is accurate only near the
resonance peaks.
0 5 10 15 20-30
-25
-20
-15
-10
-5
0
5
Frequency, Hz
LF T
ire F
orce
/ LF
Sho
ck F
orce
, dB
IdealEstimate
0 5 10 15 20-60
-50
-40
-30
-20
-10
0
LF T
ire F
orce
/ R
F S
hock
For
ce, d
B
Frequency, Hz0 5 10 15 20
-40
-35
-30
-25
-20
-15
-10
-5
LF T
ire F
orce
/ LR
Sho
ck F
orce
, dB
Frequency, Hz0 5 10 15 20
-40
-35
-30
-25
-20
-15
-10
-5
0
LF T
ire F
orce
/ R
R S
hock
For
ce, d
B
Frequency, Hz (a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz (b)
Figure 5.10. Tire force FRF estimate: (a) FRF estimate, (b) Coherence
The last output that we show is the front ride height, whose trends may be generalized to
the rear ride height. For the front shock forces, the results are generally good except at
higher frequencies, corresponding to a zero in the FRF and high frequency attenuation.
For rear shock forces, the estimate is only good near the resonance.
131
0 5 10 15 20-110
-100
-90
-80
-70
-60
Frequency, Hz
Fron
t Rid
e H
eigh
t / L
F S
hock
For
ce, d
B
IdealEstimate
0 5 10 15 20-110
-100
-90
-80
-70
-60
Fron
t Rid
e H
eigh
t / R
F S
hock
For
ce, d
B
Frequency, Hz0 5 10 15 20
-120
-110
-100
-90
-80
-70
Fron
t Rid
e H
eigh
t / L
R S
hock
For
ce, d
B
Frequency, Hz0 5 10 15 20
-120
-110
-100
-90
-80
-70
Fron
t Rid
e H
eigh
t / R
R S
hock
For
ce, d
B
Frequency, Hz (a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1C
oher
ence
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Coh
eren
ce
Frequency, Hz (b)
Figure 5.11. Ride height FRF estimate: (a) FRF estimate, (b) Coherence
These results have shown us that our frequency response estimates for shock force inputs
are good when the response levels are high, which occurs more often for direct coupling
input/output pairs that occur along a block diagonal of the frequency response matrix and
for off-diagonal entries near resonance. Estimates are reasonable when the response
levels are low. Also the coherence function provides a useful figure of merit for the
frequency response estimates, indicating where the model will be poor.
5.3.2 FRF Preconditioning
The FRF estimates described in the previous section may be processed prior to
parametric identification to provide better results. First, poor estimates observed for the
shock force input at higher frequencies may be replaced by a residual fit of the form
( )j
high N
KeH ff
φ
= (5.48)
This fit provides a 20N dB/decade rolloff and a constant phase. The form of this residual
fit is based on the assumption that the transfer function will be dominated by the highest-
order nonzero terms in the numerator and denominator for these higher frequencies.
132
A similar fit was be performed at lower frequencies, which adds a linear phase, of the
form
( )( )j bf
low N
KeH ff
φ+
= (5.49)
Each residual may be fitted by selecting two control points off each FRF where the
coherence is good. The frequency, gain, and phase at these two control points then define
the residual model constants. If desired, the control points may be manually adjusted to
provide the desired residual fit. The high frequency control points were selected to be at
20 and 30 Hz for on-diagonal input/output pairs and were at 6 and 7 Hz for off-diagonal
pairs. The low frequency residual control points were placed approximately at 1 and 4
Hz.
To further smooth the FRF estimate prior to parametric identification, the FRFs are
lowpass filtered. Filtering a complex vector may be performed analogously to filtering a
real vector, with real filters being applied to the magnitude in dB and the unwrapped
phase. A 4th order Butterworth filter was constructed to attenuate FRF variations with a
width of 0.6 Hz or less, corresponding to 18 frequency samples. This filter was then
applied to the magnitude in dB and the unwrapped phase using the MATLAB function
filtfilt to avoid frequency shift.
A comparison of the ideal FRF, the original FRF estimate, and the modified estimate is
shown in Figure 5.12. As this figure shows, the modified FRF estimate is much
smoother, making it easier to fit a low-order transfer function.
133
0 5 10 15 20-60
-50
-40
-30
-20
Frequency, HzLF S
uspe
nsio
n V
eloc
ity /
LF S
hock
For
ce, d
B
IdealFRF EstimateFRF Modif ied
0 5 10 15 20-100
-90
-80
-70
-60
-50
-40
-30
LF S
uspe
nsio
n V
eloc
ity /
RR
Sho
ck F
orce
, dB
Frequency, Hz
0 5 10 15 20-30
-25
-20
-15
-10
-5
0
5
LF T
ire F
orce
/ LF
Sho
ck F
orce
, dB
Frequency, Hz0 5 10 15 20
-60
-50
-40
-30
-20
-10
0
LF T
ire F
orce
/ R
R S
hock
For
ce, d
B
Frequency, Hz
0 5 10 15 20-120
-110
-100
-90
-80
-70
-60
-50
Fron
t Rid
e H
eigh
t / L
F S
hock
For
ce, d
B
Frequency, Hz0 5 10 15 20
-140
-120
-100
-80
-60Fr
ont R
ide
Hei
ght /
RR
Sho
ck F
orce
, dB
Frequency, Hz
Figure 5.12. Comparison of ideal FRF, FRF estimate, and modified FRF estimate
5.3.3 Parametric Identification Results
Now that the nonparametric frequency response estimates have been calculated, the next
step is to use these results to fit a parametric model to each input/output pair using the
invfreqs function. The denominator polynomial orders were increased until good
matching was achieved, resulting in denominator orders ranging from 6 to 8. The tire
force and ride height outputs have direct feedthrough from the nearby wheelloaders, so
their numerator order is set equal to the denominator order, while all other numerator
orders are set to one order lower than the denominator. A weighting function of 1 was
used between 0.5 and 10 Hz, while a weighting function of 0.5 was used from 10 to 20
Hz.
134
The results of the parametric fit created by invfreqs are compared with the ideal response
and the modified FRF estimate in Figure 5.13. As desired, all transfer function fits
closely approximate the ideal FRF where response levels are high, while the transfer
function levels are low when the response level is low.
0 5 10 15 205
10
15
20
25
30
35
Frequency, HzLF S
uspe
nsio
n V
eloc
ity /
LF S
hock
For
ce, d
B
IdealTF Estimate
0 5 10 15 20-40
-30
-20
-10
0
10
20
30
LF S
uspe
nsio
n V
eloc
ity /
RR
Sho
ck F
orce
, dB
Frequency, Hz
0 5 10 15 2035
40
45
50
55
60
65
LF T
ire F
orce
/ LF
Sho
ck F
orce
, dB
Frequency, Hz0 5 10 15 20
10
20
30
40
50
60
LF T
ire F
orce
/ R
R S
hock
For
ce, d
B
Frequency, Hz
0 5 10 15 20-50
-40
-30
-20
-10
0
10
Fron
t Rid
e H
eigh
t / L
F S
hock
For
ce, d
B
Frequency, Hz0 5 10 15 20
-70
-60
-50
-40
-30
-20
-10
0
Fron
t Rid
e H
eigh
t / R
R S
hock
For
ce, d
B
Frequency, Hz
Figure 5.13. Result of parametric identification
5.3.4 Simulation with Identified Model
The transfer function model identified in Section 5.3.3 can now be applied to perform
full-vehicle simulations using the Simulink model shown in Figure 5.14.
135
F_linear
1e-3*eye(4)* u
normalize shock input
Ysim
To Workspace
y _linear F_LPNL2
Shock Models
u_rig_hat
u_shocks_hat
y
v _shocks
Explicit SSLinear Vehicle Model
(normalized)
U
U_hat
Drivefi le
K*u
Baseline Shocks
F_LPNL2
u_shocks
u_rig_hat
u_shocks_hat
y _linear
v _shocks
Figure 5.14. Simulink model
The continuous-time transfer function model is converted to a discrete-time state space
model sampled at 100 Hz using Tustin’s bilinear transformation in the MATLAB
function c2d. The shock velocity calculated as an output of the linear model is then used
to calculate the shock force predicted by the LPNL2 shock models and the shock force
that would be generated by the baseline shock at that velocity. The difference in the
shock force from the baseline shock force is then normalized and fed back as an input to
the linear vehicle model.
Five simulations were performed with the identified model to compare with simulations
using the ideal model. The 800 second Gaussian white noise bandlimited to 30 Hz
drivefile described in section 5.1.3 was used to excite the Simulink model. One baseline
simulation was performed with 4 nonlinear dynamic baseline shock models with a
damping coefficient of 20 lb/(in/s) and four additional simulations were performed where
the baseline shock at each corner was replaced by a nonlinear dynamic shock model with
a damping coefficient of 8 lb/(in/s).
136
The results from each simulation were bandpass filtered from 1 to 20 Hz using a 4th order
Butterworth filter. A comparison of the tire force and front ride height using the ideal
model and the identified model with the baseline shocks, the left front shock changed,
and the right front shock changed is shown in Figure 5.15. These plots show that the
simulation using the ID model match the original data well. Errors in ride height are
small, while there is good matching of the tire force at the dominant frequency band of 3-
5 Hz. Peak errors in ride height occur when the signal is changing rapidly, suggesting
phase error. The largest error occurs with the left front tire force when the left front
shock is changed, where the error magnitude is roughly 25% of the signal level.
Figure 5.15. Simulation time trace comparison
The error can be analyzed in further detail in the frequency domain. Figure 5.16 shows
the power spectral density (PSD) for the time signals in Figure 5.15. For the ride height,
the error is 50 dB lower than the signal level. For the tire force, the error is more than 20
dB lower than the signal level, except for the change in left front shock from 7 to 17 Hz.
In this frequency band, the peak error is 40 dB lower than the peak signal level.
137
Figure 5.16. Simulation PSD comparison
The main purpose of this model is to accurately predict trends in response measures as
shock setups change. Since the PSDs for the ideal simulation and the ID simulation
match well, we would expect RMS response metrics to match well. This is the case, as
Figure 5.17 illustrates.
Figure 5.17. Metric comparison
5.4 Summary
This chapter has developed a method to identify a linear full-vehicle model from sensor
measurements collected during vehicle testing that allows investigation of the effect of
shock selection in simulation. To develop the method, a seven degree of freedom vehicle
state space model was constructed to generate data sets for the identification process.
The identification process can then be judged acceptable if the identified model is similar
to the known ideal model.
138
The frequency response function (FRF) of the state space model was then explored to
provide insight prior to the identification process. This analysis showed the importance
of both the aeroloaders and the wheelloaders to excite shock velocity at low and high
frequencies, respectively. It also showed frequency ranges where we might expect to
have poor coherence for certain input/output pairs due to low signal levels.
If the shock force is almost linearly dependent on the wheelloader and aeroloader inputs,
the typical FRF estimation process will fail. A modified FRF estimation process was
described which uses test data from several different shock configurations to address the
shock force dependence issue. This method provided FRF estimates, which matched the
ideal FRF where the coherence was high. When the coherence was poor due to low
shock force or output levels at low or high frequency, the FRF estimate was replaced by a
residual fit with a linear phase and linear dB/decade rolloff. The FRF estimate was
further smoothed by applying a filter, reducing FRF variations with a width of 0.6 Hz or
less.
Once an acceptable FRF estimate was calculated, a parametric model was identified by
fitting transfer functions to each input-output pair. This parametric model was then be
implemented in Simulink using the shock models described in Chapter 3. Simulations
were run for 5 different shock configurations and compared to simulation results from the
ideal model, showing that the identified model could accurately reproduce the original
response and predict trends in response metrics.
Now that this identification process has been shown to accurately reproduce the original
system behavior, it can now be applied to data collected from actual 8-post rig testing.
139
Chapter 6 8-Post Rig Testing and Identification
This chapter discusses the process of conducting an 8-post rig test, using the results to
identify a model to predict the results of future tests, and to validate the model accuracy.
First, the experimental setup for the 8-post rig tests is described. Next, the results for the
preliminary testing will be used to evaluate the validity of linear modeling for each input-
output pair, which will provide guidance during the ID process. The system
identification methods and results are then presented, followed by model validation.
6.1 Experimental Setup
The 8-post rig test in this study is comprised of four major building blocks: the 8-post rig
itself, the instrumented test vehicle installed on the rig, the drivefiles used to command
the rig, and the shock configurations installed on the vehicle. Once these building blocks
are defined, test procedures can be easily defined as needed. The following sections
detail each of these four building blocks independently then describe the actual tests that
were conducted.
6.1.1 8-Post Rig
All rig testing was conducted on the 8-post rig at the Virginia Institute for Performance
Engineering and Research (VIPER). During an 8-post rig test, the tires are supported by
4 hydraulic actuators, or wheelloaders, that simulate inputs from the track surface. Four
pneumatic actuators, or aeroloaders, attach to the vehicle’s chassis to provide heave
force, roll moment, pitch moment, and warp moment, which simulate the effect of inertial
and aerodynamic forces present during a track test. The vehicle is shaken on the rig to
140
simulate how the car would respond at a particular track or to characterize the vehicle
response to more general waveforms such as sine waves or broadband signals.
The test vehicle attached to the rig is shown in Figure 6.1a. From the 8-post lab, the view
of the components of the rig is obscured by a reconfigurable floor structure, known as a
pit cover, which allows vehicles to be driven directly onto the rig. The rig can be clearly
seen with the pit cover removed, as shown in Figure 6.1b. The rig rests on a seismic
mass, which consists of 400 tons of concrete and steel isolated from the building
foundation, ensuring that most of the actuator energy is used to accelerate the test vehicle
and not the ground underneath. A steel base plate is mounted to the seismic mass, which
is used to mount the wheelloaders and aeroloaders, as shown in Figure 6.1c. The
wheelloader is clamped to the baseplate, while the aeroloader is attached to a swivel joint
that mounts to a pedestal, which is then clamped to the baseplate.
(a) (b) (c)
Figure 6.1. 8-post rig: (a) NASCAR Cup car on rig, (b) Rig with pit cover removed,
(c) Wheelloader and aeroloaders mounted under pit cover (photos by author, 2009)
The rig is installed below the ground in a recess in the seismic mass, known as the pit.
This allows the wheel platens to be at ground level, so a vehicle can drive across the pit
cover and onto the wheel platens.
When a vehicle is installed on the rig, each tire is supported by a wheel platen, as shown
in Figure 6.2a. The wheel platen is mounted to the wheelloader, which provides the
interface between the wheelloader and the tire. On the top of the wheel platen is a white
Teflon plate, which reduces friction between the tire and the wheel platen. A low friction
tire-wheelloader interface reduces the potential for variations in the vehicle’s equilibrium
141
point due to friction and also reduces jacking forces. There are also lateral and
longitudinal restraint bars around the perimeter of the wheel platen, which prevent the
vehicle from falling off the wheel platen. These restraint bars do not touch the tire during
normal testing. The force applied by the wheelloader is measured by four triaxial load
cells, as shown in Figure 6.2b. The vertical force measured by the four load cells is
connected to a summing box to allow direct measurement of the vertical wheel platen
force.
(a) (b) (c)
Figure 6.2. Rig attachments to vehicle: (a) Wheel platen and wheel restraint,
(b) Wheel platen load cells, (c) Aeroloader mounting and load cell
(photos by author, 2009)
Four pneumatic actuators, or aeroloaders, are also attached to the vehicle’s frame, as
shown in Figure 6.2c. The aeroloader is attached at the bottom using a swivel joint and at
the top using a rod end. These joints ideally produce free-free boundary conditions
across the aeroloader, which would make the aeroloader an ideal two-force member that
provides only axial force along the line between the two pivots. A blue load cell is
mounted on the aeroloader shaft to measure the applied force, as shown in Figure 6.2c.
Each wheelloader consists of a hydraulic actuator which has a stroke of 12.4 inches, a
control bandwidth of 125 Hz, can produce 7,400 lb of dynamic force, and travels at
speeds up to 180 in/s. Each actuator includes a Macro Sensors linear variable differential
transformer (LVDT) for position measurement, a PCB accelerometer, and four Michigan
Scientific triaxial loadcells to measure the three forces and three moments transferred to
the wheel platen. The four wheelloaders are powered by a 200 HP, 100 GPM, 3000 psi
hydraulic pump with over 30 gallons of accumulation to prevent transient pressure drops.
142
Each aeroloader consists of a pneumatic actuator which has a stroke of 24 inches, a
control bandwidth of 9 Hz, can produce 4300 lb of force, and control at velocities up to
12 in/s. Each actuator includes a Futek load cell and an Interface LVDT. The four
aeroloaders are powered by a 50 HP air compressor, which provides refrigerator-dried air
at a working pressure of 120 psi.
The 8-post rig is controlled by the Moog-FCS SmarTest controller shown in Figure 6.3a.
The controller receives the analog sensor signals from the rig and converts them to a
digital signal, which is then available to the controller’s real-time processor. The real-
time processor then checks to see if any safety limits have been exceeded requiring rig
shutdown. If no safety limits have been tripped, the processor applies a control policy to
calculate the actuator command signals, which are then converted to an analog signal
through an A/D and routed to the actuator control valves. The controller also includes 24
additional channels of analog input and data logging capability.
(a)
(b)
Figure 6.3. Test control: (a) SmarTest controller, (b) FasTest PC interface
(photos by author, 2009)
Test management is performed using Moog-FCS’s FasTest software on a PC networked
to the controller as shown in Figure 6.3b. The software allows the user to define a
sequence of instructions that define the actuator commands or drivefile, data acquisition,
and safety limits. After a sequence of instructions has been constructed, the sequence can
be downloaded to the real-time processor for execution.
143
For each test, we used a sequence of instructions that played a drivefile and recorded the
vehicle and rig sensor measurements. The vehicle instrumentation and drivefiles that
were used are described in the next two sections.
6.1.2 Test Vehicle Setup
The test vehicle used for the 8-post rig tests, as shown in Figure 6.1a, is a 2007 NASCAR
Cup car donated to VIPER by Petty Enterprises. This car’s suspension consists of an
independent short-long arm (SLA) front suspension and a solid-axle trailing-arm rear
suspension, as shown in Figure 6.4. At all four corners, there is a coil spring and a
Penske 7300 shock absorber installed. In the front, there is also an anti-roll bar, which
provides some coupling across the front axle.
(a) (b)
Figure 6.4. Test vehicle suspension: (a) Independent SLA front suspension,
(b) Solid-axle trailing-arm rear suspension (photos by author, 2009)
In addition to the sensors already present on the 8-post rig actuators, instrumentation was
installed throughout the vehicle to characterize the suspension performance. Each shock
has a Penny and Giles shock potentiometer mounted across it with a motion ratio close to
1 with respect to the shock travel. The right front shock potentiometer is shown in
Figures 6.5a. During the identification tests, shock force will be measured using the
144
Penske strain gauge eyelet and Beru F1 inline amplifier shown in Figure 6.5b. The strain
gauge eyelet threads onto the shock shaft, replacing the stock eyelet.
PCB accelerometers were also used near the suspension and in the cockpit, as shown in
Figure 6.6. Accelerometers were used on each corner to measure the wheel acceleration
and the frame acceleration above the suspension. A triaxial accelerometer was used to
measure acceleration near the vehicle center of gravity and a single axis accelerometer
was used to measure acceleration near the driver’s torso, which may be unfavorable to
driver comfort. All accelerometers were conditioned using a 16 channel PCB signal
conditioner with programmable gain and 8th order elliptical filter, which was
programmed to give the accelerometers a sensitivity of approximately 1 V/g and a cutoff
frequency of 250 Hz.
(a)
(b)
Figure 6.5. Shock instrumentation: (a) Shock potentiometer, (b) Shock load cell
(photos by author, 2009)
145
(a) (b)
Figure 6.6. Accelerometers: (a) Wheel and chassis accelerometers,
(b) Cockpit accelerometers (photos by author, 2009)
In addition to the measured channels described above, additional math channels are
calculated during post-processing that will be used for analysis. A summary of all rig
channels, vehicle sensor channels, and math channels is shown in Table 6.1. Calculation
of math channels is described below.
Table 6.1. Summary of measured and calculated channels
24 Rig Signals1-4 Wheelloader Position5-8 Aeroloader Position9-12 Wheelloader Force13-16 Aeroloader Force17-20 Wheelloader Command21-24 Aeroloader Command
24 Vehicle Sensors25-28 Shock Potentiometer29-32 Shock Force33-36 Wheel Accelerometer37-40 Chassis Accelerometer41-43 CG Triaxial Accelerometer
44 Driver Accelerometer
18 Math Channels45-48 Aeroloader Heave/Pitch/Roll/Warp Position49-52 Aeroloader Heave/Pitch/Roll/Warp Force53-56 Aeroloader Heave/Pitch/Roll/Warp Command57-58 Front and Rear Ride Heights59-62 Shock Velocity
146
It is often desirable to describe the aeroloader position and force in terms of heave, pitch,
roll, and warp modes instead of on a single actuator basis. Assuming left-right aeroloader
symmetry, the transformation between aeroloader displacement and mode displacement
is
' ' ' '2 ' 2 ' 2 ' 2 '
1 1 1 12 ' 2 ' 2 ' 2 '
' ' ' '' ' ' ' ' ' ' '
1 1 1 1
Heave DisplacementPitch AngleRoll AngleWarp Angle
f f r r
f f r r
m x
b b a aL L L L
L L L Lb b a a
L t L t L t L t
t t t t
X T X
− −
− −
− −
=
=
''''
LF
RF
LR
RR
zzzz
(6.1)
where Xm is the vector of mode displacements, X is the vector of actuator displacements,
and Tx is the transformation matrix. The constants a’, b’, L’, tf’, tr’ define the aeroloader
geometry about a reference point along the vehicle centerline. Roll angle is defined to be
the roll angle at the reference point, assuming roll angle varies linearly along the length
of the chassis. Warp angle is defined to be the difference between front and rear roll
angles. Since the aeroloaders attach to the chassis, these math channels also describe the
chassis heave, pitch, roll, and warp. Similarly, the transformation between aeroloader
forces and aeroloader mode forces can be written
' ' ' '2 2 2 2
' ' ' '2 2 2 2
1 1 1 1 'Heave Force' ' ' ' 'Pitch Moment
'Roll Moment'Warp Moment
f f r r
f f r r
m f
LF
RFt t t t
LRt t t t
RR
F T F
Fa a b b F
FF
− −
− −
=
− − =
(6.2)
where warp moment is defined to be the difference between the front and rear roll
moments. Since the aeroloaders will be run in force control for this study, the
transformation between aeroloader commands and mode commands can be written using
the transformation matrix Tf.
147
Another important measurement for aerodynamic performance is the chassis ride height
at the front valence and rear spoiler height. Deviation from an optimal chassis position
may reduce total aerodynamic downforce, increase drag, or disturb the aerodynamic
downforce distribution. The front and rear ride heights can be easily calculated using the
chassis heave and pitch.
Since our shock model requires shock velocity as an input, one key element of our
vehicle model is to predict the shock velocity. We estimate the shock velocity by
numerically differentiating the lowpass filtered shock potentiometer measurement using
MATLAB’s gradient function. This function uses a central difference approximation to
calculate the derivative.
6.1.3 Drivefiles
Several different drivefiles were used during 8-post rig testing, for various purposes
including better understanding the vehicle and acceptable excitation levels, identifying
the vehicle model, model validation, and suspension evaluation.
The first drivefiles tested were a series of sine tests. Sine tests were performed to
determine acceptable excitation levels at different frequencies and to provide model
validation data. The wheelloaders were commanded sine wave positions in heave, pitch,
and roll with frequencies chosen to capture the dynamics before, during, and after
resonance of that particular mode.
Table 6.2. Sine test frequencies and amplitudes
Frequency Amplitude Frequency AmplitudeHz in Hz in1 1 1 12 1 2 0.74 0.7 3 0.56 0.5 5 0.58 0.3 6 0.310 0.315 0.120 0.05
Heave Pitch and Roll
148
After sine testing was completed, a random drivefile was designed for use in system
identification. Each actuator command within the drivefile was constructed by first
generating a Gaussian time sequence, filtering it to achieve a desired spectrum shape, and
then adjusting the scale and mean value to desired values. To perform both multiple-
input and single-input frequency response estimation, an 800 second drivefile with all 8
actuators active and 8 100 second drivefiles with only one actuator active in each
drivefile were created. Inactive aeroloaders were commanded with a DC downforce that
provided approximately 3 inches of travel at the front aeroloaders and 2.5 inches of travel
at the rear aeroloaders, while inactive wheelloaders were commanded zero displacement.
The relative amplitude spectra for the aeroloaders and the wheelloaders are shown in
Table 6.3. The aeroloader spectrum is 100% up to 3 Hz, 10% at 4 Hz, 1% at 5 Hz, and
zero above 5 Hz due to the aeroloader bandwidth of 5 Hz. The wheelloader spectrum is
similar to the heave sine limits found during heave sine testing, and is similar to the
random spectrum used for shock testing and quarter-car testing. These spectra were used
to fit a digital filter with similar frequency response using the Yule-Walker method. The
filter was then applied to the Gaussian time sequence to provide approximately the
desired spectra.
Table 6.3. Random signal relative amplitude spectrum
Frequency Amplitude Frequency AmplitudeHz in Hz in0 1 0 1
0.5 1 0.5 13 1 1 14 0.1 2 0.85 0.01 3 0.7
5 0.58 0.310 0.315 0.320 0.130 0.01
Aeroloader Wheelloader
The signals were then adjusted to have the desired mean value and scale. The
wheelloaders were adjusted to zero mean and a peak value of 0.4 inches, providing an
149
RMS of approximately 0.11 inches. The mean aeroloader force commands were adjusted
to have approximately 3 inches of deflection in the front and 2.5 inches of deflection in
the rear. The aeroloader scale was set to a peak variation from the mean force of 560 lb,
which provided aeroloader force commands of approximately 160 lb RMS, not including
the mean downforce.
The next drivefile used was a 50 second, 0.4 inch peak-to-peak square wave with a period
of 10 seconds to all four wheelloaders, while the aeroloaders were commanded a DC
downforce. This drivefile will be used to evaluate the transient behavior of the vehicle,
determine the rate of shock absorber recovery, and to validate the model for transient
events.
The next drivefile used consists of a series of static aeroloader force commands, which
will be used for spring rating the vehicle and providing a model at DC. The aeroloader
forces are varied from zero to a maximum force determined to be near maximum
suspension travel. The four aeroloader force commands are varied from zero force to
max force and held constant at 5 different forces, as shown in Figure 6.7. This gives
force-deflection data during motion dominated by chassis heave. The process is repeated
moving from max force to zero force to determine hysteresis. Additional tests were run
in a similar manner, as shown in Figure 6.6 to excite chassis pitch, roll, and warp. Single
actuator tests were also run to test single-corner loading.
150
0 20 40 60 80 100 120-1000
-800
-600
-400
-200
0Heave
Aer
oloa
der F
orce
, lb
time, seconds
LFRFLRRR
0 20 40 60 80 100 120-1000
-800
-600
-400
-200
0Pitch
Aer
oloa
der F
orce
, lb
time, seconds
0 20 40 60 80 100 120-1000
-800
-600
-400
-200
0Roll
Aer
oloa
der F
orce
, lb
time, seconds0 20 40 60 80 100 120
-1000
-800
-600
-400
-200
0Warp
Aer
oloa
der F
orce
, lb
time, seconds
Figure 6.7. Chassis spring rating tests
The last drivefile used is a drivefile that was created using drivefile iteration to represent
a lap in a NASCAR Cup car at Richmond International Raceway. This type of track-
based drivefile is what a race team would typically use to evaluate their vehicle’s
performance for a particular track. In this study, we will use it both to evaluate the
quality of different shock selections and to validate our model’s ability to predict quality
of different shock selections.
6.1.4 Shock Builds
This section describes the shock builds that will be used for system identification and
model validation. First, a baseline shock build with a damping coefficient of
approximately 20 lb/(in/s) was chosen. Four of these baseline shocks were built and
installed on the car as a baseline setup. During the identification process, shocks must be
changed to separate the response due to shock force from the response due to actuator
inputs. To ensure that the change in shock force and change in sensor response was
significant, a shock build with a damping coefficient of approximately 9 lb/(in/s) was
chosen.
151
The four baseline builds and the alternate build were tested and modeled using the
procedures detailed in Chapter 3. The force-velocity plots for these shocks are shown in
Figure 6.8.
-20 -15 -10 -5 0 5 10 15 20 25-500
-400
-300
-200
-100
0
100
200
300
400
500
velocity, in/s
Forc
e, lb
s
-20 -15 -10 -5 0 5 10 15 20 25-500
-400
-300
-200
-100
0
100
200
300
400
500
velocity, in/s
Forc
e, lb
s
(a)
-20 -15 -10 -5 0 5 10 15 20 25-500
-400
-300
-200
-100
0
100
200
300
400
500
velocity, in/sFo
rce,
lbs
-20 -15 -10 -5 0 5 10 15 20 25-500
-400
-300
-200
-100
0
100
200
300
400
500
velocity, in/sFo
rce,
lbs
(b)
Figure 6.8. Shock builds: (a) Baseline, (b) Alternate
6.1.5 Experimental Procedure
The testing that was performed for this chapter will be used to identify a model of a
vehicle on the rig and to validate that model. Now that the building blocks for
conducting an 8-post rig test have been described, the experimental procedure may be
concisely defined. During each test, data was collected at 1250 Hz using the sensors
described in Section 6.1.2. Unless otherwise stated, the baseline shock builds A-D were
used for testing.
Identification tests were performed using the random excitation described in Section
6.1.3. First, a baseline test was run with the 800 second 8-input random drivefile and the
baseline shocks installed. The data from the baseline test will allow us to model the
relationship between the actuators and the sensors. Next, an alternate test is run with the
same drivefile and the left front shock replaced with the alternate shock build. The
alternate shock build provides about half the damping that the baseline shock provides,
which provides a significant change in the shock force. The data collected from the
alternate test along with data from the baseline test will allow us to model the relationship
152
between shock force and sensor response. Alternate tests are repeated for each corner,
where the baseline shock for that corner is replaced by the alternate shock, while all other
shocks are the baseline shocks. Once all four alternate tests are completed, we can
determine the influence of all four shock forces on sensor response.
The baseline identification tests are repeated using a series of single-input tests to allow
us to analyze the behavior of each input/output pair in further detail. Eight tests are
performed using the baseline setup tests where one actuator is excited at a time using a
100 second random drivefile. This will provide a clear picture of how actuator input
energy affects the sensor response, which will aid in model development.
To validate the model’s ability to predict the response to a drivefile representing a
particular test track, the vehicle was excited in the baseline shock configuration using the
Richmond International Raceway track drivefile. To validate the model’s ability to
predict the change in response due to shock selection, the drivefile was repeated while
each shock was replaced by the alternate shock.
Further validation tests were performed on the baseline setup including sine tests, bump
tests, and static tests. These will allow us to evaluate the model’s ability to predict the
vehicle response for different drivefiles to steady-state inputs at different frequencies, to
transient inputs, and to static inputs.
6.2 Linear Model Validity
This section evaluates the validity of a linear model to approximate each input-output
pair. For the wheelloader and aeroloader inputs, we consider the results for the one input
at a time tests for the linear baseline setup. For the shock inputs, we consider the change
in response from the baseline linear response for the multiple-input random test when one
shock at a time is replaced by an alternate shock. These experiments give us valuable
information about how each input contributes to each output and if the input-output pair
can be accurately modeled using a linear model.
153
For this analysis, we will consider the combined importance of response energy and
coherence in linear system identification, as summarized in Table 6.4. Ideally, the
coherence will be close to 1 for each input-output pair across the frequency range of
interest. This means the input-output relationship observed in the data is accurately
described by a linear relationship and it is reasonable to assume that a linear model is
appropriate.
Table 6.4. Importance of coherence and output energy for linear modeling
High LowCondition I Condition III
Good linear estimate Poor linear estimate
Model predicts response Cannot predict response
Condition II Condition IVGood linear estimate Poor linear estimate
Model predicts low response We can choose low response
Res
pons
e En
ergy
High
Low
Coherence
Unfortunately, poor coherence may be caused by various factors including poor signal-to-
noise ratio or system nonlinearity. If the coherence is low, for example below 0.5, a
linear model only describes 50% or less of the response energy observed in the data.
While this is not ideal, the coherence, or relative amount of the response energy described
by a linear model, does not fully describe the potential accuracy of a linear system model,
and can be a bit misleading. If coherence is poor and the response energy is high, then
there will be significant amount of absolute error in the output energy predicted by a
linear model. If the response energy is small or negligible, however, a poor coherence
does not indicate a large absolute error in the output energy predicted by a linear model.
When the output energy is small, we only need to ensure that the predicted response
levels are small or zero, and there will be small absolute error.
For brevity, this section only displays the results for the shock velocity, tire force at the
right front and left rear locations. Also shown are the results for front and rear ride
height. Similar trends have been found for the other sensor outputs and locations.
154
Figure 6.9 shows the results for the right front and left rear shock velocity resulting from
wheelloader inputs. Figure 6.9a shows two seconds of the time response as one
wheelloader is applied at a time. For the right front shock velocity, the only time the
response was significant was when the nearby right front wheelloader was excited. For
the left rear shock velocity, the most significant response was due to the nearby left rear
wheelloader, with a smaller but significant response due to the right rear wheelloader.
This indicates that the shock velocity in the independent front suspension is only
significantly excited by the closest wheelloader, while for the solid axle rear suspension
the shock velocity is heavily influenced by the nearby wheelloader with additional
significant contribution from the opposite side of the solid axle.
51 51.2 51.4 51.6 51.8 52-15
-10
-5
0
5
10
15RF Shock Velocity due to Wheelloader Command
time, seconds
RF
Sho
ck V
eloc
ity, i
n/s
LF InputRF InputLR InputRR Input
51 51.2 51.4 51.6 51.8 52-10
-5
0
5
10
15LR Shock Velocity due to Wheelloader Command
time, seconds
LR S
hock
Vel
ocity
, in/
s
(a)(a)(a)(a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1RF Shock Velocity / Wheelloader Command
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 5 10 15 200
0.2
0.4
0.6
0.8
1LR Shock Velocity / Wheelloader Command
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 5 10 15 20-20
-10
0
10
20
30
40RF Shock Velocity / Wheelloader Command
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 5 10 15 20-20
-10
0
10
20
30
40LR Shock Velocity / Wheelloader Command
Mag
nitu
de, d
B
Frequency, Hz
(b)(b)(b)(b)
Figure 6.9. Shock velocity due to wheelloader input: (a) Time response,
(b) FRF magnitude, (c) Coherence
This result is also evident when looking at the FRF magnitude in Figure 6.9b. For the
right front shock velocity, the FRF for the closest wheelloader is 20-30 dB higher than
the other wheelloader inputs over most of the frequency band. For the left rear shock
velocity, the FRF is 20-30 dB lower for the front inputs and about 10 dB lower for the
right rear input compared to the nearby left rear wheelloader input. These observations
clearly indicate that the shock velocity response level is only significant in the front due
to the closest wheelloader, while in the back the response is only significant for the rear
wheelloaders. Also note that the magnitude response drops toward zero at low
155
frequency, since no shock velocity is excited at low frequency as the wheels move with
the chassis in a rigid body mode.
The coherence for these outputs is shown in Figure 6.9c. As expected, the coherence for
the nearby wheelloader is high, except at low frequency where there is little shock
velocity. The coherence is also high for the left rear shock velocity for the right rear
wheelloader input except at a dip in the magnitude near 7 Hz. The coherence for the
remaining input-output pairs ranges from good to poor.
A key observation from this plot is that at frequencies where an input-output pair
produces a significant response level, the coherence is good. For input-output pairs
where the response levels are low, the transfer function may be approximated as zero
with minimal error.
Figure 6.10 shows the shock velocity due to aeroloader inputs. The coherence is
generally high for all inputs, with the possible exception of the right front wheelloader
input for the left rear shock velocity. For this input-output pair, the coherence is low
when the response level is low.
51 51.2 51.4 51.6 51.8 52-5
0
5RF Shock Velocity due to Aeroloader Command
time, seconds
RF
Sho
ck V
eloc
ity, i
n/s
LF InputRF InputLR InputRR Input
51 51.2 51.4 51.6 51.8 52-3
-2
-1
0
1
2
3
4LR Shock Velocity due to Aeroloader Command
time, seconds
LR S
hock
Vel
ocity
, in/
s
(a)(a)(a)(a)
0 1 2 3 4 50.2
0.4
0.6
0.8
1RF Shock Velocity / Aeroloader Command
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1LR Shock Velocity / Aeroloader Command
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 1 2 3 4 5-60
-55
-50
-45
-40
-35RF Shock Velocity / Aeroloader Command
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 1 2 3 4 5-70
-65
-60
-55
-50
-45
-40
-35LR Shock Velocity / Aeroloader Command
Mag
nitu
de, d
B
Frequency, Hz
(b)(b)(b)(b)
Figure 6.10. Shock velocity due to aeroloader input: (a) Time response,
(b) FRF magnitude, (c) Coherence
Next, we consider the effect of wheelloader inputs on tire force in Figure 6.11. For each
wheelloader force, the most significant response is due to the nearby wheelloader, while
156
the rear wheelloader force has an additional significant contribution due to the
wheelloader across the rear axle. The coherence is good for the nearby wheelloader input
for the front shock velocity, and for both rear inputs for the rear shock velocity.
51 51.2 51.4 51.6 51.8 521000
1200
1400
1600
1800
2000
2200RF Wheelloader Force due to Wheelloader Command
time, seconds
RF
Whe
ello
ader
For
ce, l
bs
LF InputRF InputLR InputRR Input
51 51.2 51.4 51.6 51.8 52600
800
1000
1200
1400
1600
1800LR Wheelloader Force due to Wheelloader Command
time, seconds
LR W
heel
load
er F
orce
, lbs
(a)(a)(a)(a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1RF Wheelloader Force / Wheelloader Command
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 5 10 15 200
0.2
0.4
0.6
0.8
1LR Wheelloader Force / Wheelloader Command
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 5 10 15 2030
40
50
60
70
80RF Wheelloader Force / Wheelloader Command
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 5 10 15 2010
20
30
40
50
60
70LR Wheelloader Force / Wheelloader Command
Mag
nitu
de, d
B
Frequency, Hz
(b)(b)(b)(b)
Figure 6.11. Tire force due to wheelloader input: (a) Time response,
(b) FRF magnitude, (c) Coherence
The results for tire force due to aeroloader force are shown in Figure 6.12. The
coherence is generally good, except for the front aeroloaders for the tire forces on the
opposite side of the car. The influence of aeroloader force on tire force can be better
understood by considering the aeroloader configuration and the suspension load paths,
illustrated in Figure 6.13. The aeroloaders are located laterally within the tire contact
patch, while they are located close to the tires but between the axles longitudinally. Since
the aeroloaders are close to the tires, the closest tire reacts most of the aeroloader force.
Since the aeroloaders are offset from the tires in the longitudinal direction, a small
amount of the aeroloader force will be supported by the distant axle. Since the
aeroloaders have limited lateral offset from the tires, however, it might be expected that
very little aeroloader force would be transmitted from one side of the car to the other.
This would be true if all four corners had a fully independent suspension where the
primary suspension stiffness could be represented directly above the tire. In the case of
the solid rear axle, the springs that react to the aeroloader force are connected to the solid
axle inboard of the tires. Since the spring is inboard, when a rear aeroloader pulls down
on the car, it also causes a moment about the nearest spring, causing the opposite side of
157
the car to rise. This causes transmission of the rear aeroloader force to the opposite side
of the car. In the case of the independent front suspension, the moment described above
is reacted by the control arms where they attach to the chassis.
51 51.2 51.4 51.6 51.8 521100
1200
1300
1400
1500
1600RF Wheelloader Force due to Aeroloader Command
time, seconds
RF
Whe
ello
ader
For
ce, l
bs
LF InputRF InputLR InputRR Input
51 51.2 51.4 51.6 51.8 521100
1150
1200
1250
1300
1350
1400
1450LR Wheelloader Force due to Aeroloader Command
time, seconds
LR W
heel
load
er F
orce
, lbs
(a)(a)(a)(a)
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1RF Wheelloader Force / Aeroloader Command
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1LR Wheelloader Force / Aeroloader Command
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 1 2 3 4 5-25
-20
-15
-10
-5
0RF Wheelloader Force / Aeroloader Command
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 1 2 3 4 5-40
-30
-20
-10
0LR Wheelloader Force / Aeroloader Command
Mag
nitu
de, d
B
Frequency, Hz
(b)(b)(b)(b)
51 51.2 51.4 51.6 51.8 521100
1200
1300
1400
1500
1600RF Wheelloader Force due to Aeroloader Command
time, seconds
RF
Whe
ello
ader
For
ce, l
bs
LF InputRF InputLR InputRR Input
51 51.2 51.4 51.6 51.8 521100
1150
1200
1250
1300
1350
1400
1450LR Wheelloader Force due to Aeroloader Command
time, seconds
LR W
heel
load
er F
orce
, lbs
(a)(a)(a)(a)
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1RF Wheelloader Force / Aeroloader Command
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1LR Wheelloader Force / Aeroloader Command
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 1 2 3 4 5-25
-20
-15
-10
-5
0RF Wheelloader Force / Aeroloader Command
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 1 2 3 4 5-40
-30
-20
-10
0LR Wheelloader Force / Aeroloader Command
Mag
nitu
de, d
B
Frequency, Hz
(b)(b)(b)(b)
Figure 6.12. Tire force due to aeroloader input: (a) Time response,
(b) FRF magnitude, (c) Coherence
Aero 1
Aero 2
Aero 3
Aero 4
LR Spring
RR Spring
Aero 1
Aero 2
Aero 3
Aero 4
LR Spring
RR Spring
(a) LR Tire Force RR Tire Force
LR Aero Force RR Aero Force
LR Tire Force RR Tire Force
LR Aero Force RR Aero Force
(b)
Figure 6.13. (a) Aeroloader and rear spring locations, (b) Simplified rear axle model
Next, we consider the effect of wheelloaders on ride height, as shown in Figure 6.14.
Ride height is defined as the vertical distance from a chosen point on the chassis to a
point directly below the chassis point on a virtual ground plane defined by the
wheelloader positions. Front and rear ride heights are calculated using chassis points
along the vehicle centerline at the nose and spoiler, respectively. Like shock velocity, the
wheelloaders do not change the ride height at low frequency, as the chassis moves like a
rigid body. The coherence is good for the wheelloaders for the axle closest to the ride
158
height measurement, which also causes the highest response levels. The wheelloader
inputs for the opposite axle also have good coherence, except at low frequency, which
corresponds to low response levels.
51 51.2 51.4 51.6 51.8 52-16.3
-16.2
-16.1
-16
-15.9
-15.8Front Ride Height due to Wheelloader Command
time, seconds
Fron
t Rid
e H
eigh
t, in
LF InputRF InputLR InputRR Input
51 51.2 51.4 51.6 51.8 52-9.3
-9.2
-9.1
-9
-8.9
-8.8Rear Ride Height due to Wheelloader Command
time, seconds
Rea
r Rid
e H
eigh
t, in
(a)(a)(a)(a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1Front Ride Height / Wheelloader Command
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 5 10 15 200
0.2
0.4
0.6
0.8
1Rear Ride Height / Wheelloader Command
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 5 10 15 20-30
-25
-20
-15
-10
-5
0Front Ride Height / Wheelloader Command
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 5 10 15 20-25
-20
-15
-10
-5
0
5Rear Ride Height / Wheelloader Command
Mag
nitu
de, d
B
Frequency, Hz
(b)(b)(b)(b)
Figure 6.14. Ride height due to wheelloader input: (a) Time response,
(b) FRF magnitude, (c) Coherence
Next, we consider the ride height due to aeroloader inputs in Figure 6.15. The response
levels are highest for the aeroloaders closest to the ride height measurement. The
coherence is generally good, with the exception of the influence of front aeroloaders on
rear ride height, which corresponds to low response levels.
159
51 51.2 51.4 51.6 51.8 52-16.4
-16.2
-16
-15.8
-15.6
-15.4Front Ride Height due to Aeroloader Command
time, seconds
Fron
t Rid
e H
eigh
t, in
LF InputRF InputLR InputRR Input
51 51.2 51.4 51.6 51.8 52-9.3
-9.2
-9.1
-9
-8.9
-8.8
-8.7
-8.6Rear Ride Height due to Aeroloader Command
time, seconds
Rea
r Rid
e H
eigh
t, in
(a)(a)(a)(a)
0 1 2 3 4 5
0.4
0.5
0.6
0.7
0.8
0.9
1Front Ride Height / Aeroloader Command
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1Rear Ride Height / Aeroloader Command
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 1 2 3 4 5-75
-70
-65
-60
-55
-50Front Ride Height / Aeroloader Command
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 1 2 3 4 5-100
-90
-80
-70
-60
-50Rear Ride Height / Aeroloader Command
Mag
nitu
de, d
BFrequency, Hz
(b)(b)(b)(b)
Figure 6.15. Ride height due to aeroloader input: (a) Time response,
(b) FRF magnitude, (c) Coherence
Next, we consider the shock force inputs. Since we do not directly excite shock force, it
is harder to estimate the single-input response. To estimate a single-input response, we
consider the change in response from the baseline linear response for the multiple input
random tests when one shock at a time is replaced by an alternate shock. The change in
output, change in shock force, and FRF are calculated using the procedure described in
Chapter 5. If all the baseline shocks are linear, this should give us a single-input FRF
estimate and coherence. In reality, all the shocks are slightly nonlinear, and this small
nonlinearity will act as additional inputs that are not accounted for in the single input
model, slightly reducing coherence and FRF accuracy. Due to this, we will only use the
procedure described in Chapter 5 to give us a qualitative estimate of the validity of the
linear model for various input-output pairs, and we will adopt a different FRF estimation
strategy which will be described in Section 6.3.1 for system identification.
Figure 6.16 shows the results for change in shock velocity due to change in shock force.
The influence of changing the shock force on shock velocity is most evident at the corner
the shock force is changed. The coherence for this collocated input-output pair is high,
except near an anti-resonance at 4 Hz. The rear shock velocity coherence is also high at
all frequencies except for the anti-resonance for both rear shock force inputs. The
coherence is good at low frequencies for all input-output pairs, except for the shock
160
velocity due to shock force across the diagonals of the car, which also corresponds to low
output levels.
100 100.2 100.4 100.6 100.8 101-8
-6
-4
-2
0
2
4
6RF Shock Velocity due to Shock Force
time, seconds
RF
Sho
ck V
eloc
ity, i
n/s
LF InputRF InputLR InputRR Input
100 100.2 100.4 100.6 100.8 101-5
0
5LR Shock Velocity due to Shock Force
time, seconds
LR S
hock
Vel
ocity
, in/
s
(a)(a)(a)(a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1RF Shock Velocity / Shock Force
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 5 10 15 200
0.2
0.4
0.6
0.8
1LR Shock Velocity / Shock Force
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 5 10 15 20-90
-80
-70
-60
-50
-40
-30
-20RF Shock Velocity / Shock Force
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 5 10 15 20-100
-80
-60
-40
-20LR Shock Velocity / Shock Force
Mag
nitu
de, d
B
Frequency, Hz
(b)(b)(b)(b)
Figure 6.16. Shock velocity due to shock force input: (a) Change in time response,
(b) FRF magnitude, (c) Coherence
Next, we consider the influence of shock force on wheelloader force in Figure 6.17. The
coherence is good for tire force closest to the shock force and for rear tire force due to
rear shock force. For the front shock force influence on the opposite side of the car, the
coherence is low despite significant output energy, especially for the rear wheelloader
force. This is important to note because this is the first time we have observed an input-
output pair where the coherence was poor at all frequencies but the response level was
still significant. For the remaining inputs, the coherence is low, except from 2-4 Hz and
8-14 Hz, corresponding to peaks in the FRFs.
161
100 100.2 100.4 100.6 100.8 101-300
-200
-100
0
100
200
300RF Wheelloader Force due to Shock Force
time, seconds
RF
Whe
ello
ader
For
ce, l
b
LF InputRF InputLR InputRR Input
100 100.2 100.4 100.6 100.8 101-200
-150
-100
-50
0
50
100
150LR Wheelloader Force due to Shock Force
time, seconds
LR W
heel
load
er F
orce
, lb
(a)(a)(a)(a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1RF Wheelloader Force / Shock Force
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 5 10 15 200
0.2
0.4
0.6
0.8
1LR Wheelloader Force / Shock Force
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 5 10 15 20-80
-60
-40
-20
0
20RF Wheelloader Force / Shock Force
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 5 10 15 20-60
-50
-40
-30
-20
-10
0
10LR Wheelloader Force / Shock Force
Mag
nitu
de, d
BFrequency, Hz
(b)(b)(b)(b)
Figure 6.17. Tire force due to shock force input: (a) Change in time response,
(b) FRF magnitude, (c) Coherence
The last set of results we consider is the ride height due to shock force, shown in Figure
6.18. The coherence for front ride height is good for all inputs from 1-4 Hz. Above 4
Hz, the FRF magnitude is below -70 dB, meaning 100 lb change in shock force gives less
than 0.03 inches change in ride height. For the rear ride height, the results are similar.
100 100.2 100.4 100.6 100.8 101-0.6
-0.4
-0.2
0
0.2Front Ride Height due to Shock Force
time, seconds
Fron
t Rid
e H
eigh
t, in
LF InputRF InputLR InputRR Input
100 100.2 100.4 100.6 100.8 1010.1
0.2
0.3
0.4
0.5
0.6Rear Ride Height due to Shock Force
time, seconds
Rea
r Rid
e H
eigh
t, in
(a)(a)(a)(a)
0 5 10 15 200
0.2
0.4
0.6
0.8
1Front Ride Height / Shock Force
Frequency, Hz
Coh
eren
ce
LF InputRF InputLR InputRR Input
0 5 10 15 200
0.2
0.4
0.6
0.8
1Rear Ride Height / Shock Force
Frequency, Hz
Coh
eren
ce
(c)(c)(c)(c)
0 5 10 15 20-110
-100
-90
-80
-70
-60
-50
-40Front Ride Height / Shock Force
Mag
nitu
de, d
B
Frequency, Hz
LF InputRF InputLR InputRR Input
0 5 10 15 20-110
-100
-90
-80
-70
-60
-50Rear Ride Height / Shock Force
Mag
nitu
de, d
B
Frequency, Hz
(b)(b)(b)(b)
Figure 6.18. Ride height due to shock force input: (a) Changer in time response,
(b) FRF magnitude, (c) Coherence
This section has reviewed the accuracy of linear modeling for each input-output pair in
terms of response level and coherence. This analysis has shown that aside from a few
162
exceptions, coherence is high when the response levels are high. This indicates that a
linear model that has a frequency response similar to the FRF estimates where the
coherence is good and has a low frequency response where the coherence is low should
capture a large portion of the output energy. Other trends worth noting include that more
energy was transmitted for input-output pairs located within the same corner or across the
solid rear axle. This increased coupling also led to the best coherence levels. This
information can now be applied in the system identification process to address frequency
ranges where the FRF estimate is poor.
6.3 8-Post Rig System Identification
This section describes the methods and results for 8-post system identification. First, we
describe the FRF estimation method used. Next, the results for FRF estimation are
shown. Finally, the FRF estimate is used to fit transfer functions, which will be used to
simulate the results of future rig tests.
6.3.1 FRF Estimation Method
In Chapter 5, we developed a FRF estimation method for wheelloader, aeroloader, and
shock force inputs. It was shown that when the shocks are linear, the shock force is
linearly dependent on the wheelloader and aeroloader inputs, making it impossible to
identify the FRF. This issue was addressed by first performing a baseline experiment
with 4 linear shocks, then performing 4 additional experiments, where one shock at a
time was replaced by a second shock. This method allowed the FRF to be accurately
identified, with the FRF estimate matching the known theoretical result.
During simulation, we can choose our shocks to be exactly linear. In practice, all shocks
will exhibit some degree of nonlinearity. If we assume our shocks are linear, any
variation from this linear model will cause errors in the FRF estimation and the resulting
model. To accommodate realistic shock performance, the FRF estimation method is
extended to accommodate arbitrary shock forces.
163
Start with the frequency response model
rig rig shocks shocksy H u H u= + (6.3)
Assuming that most of the system energy dissipation is caused by the shocks, the FRF
matrices rigH and shocksH represent a lightly damped system. This undamped system
representation has a slowly-decaying impulse response, causing leakage errors when the
data blocks are windowed out of a larger data file. The damping of the identified system
may be arbitrarily increased by redistributing the energy dissipated by the shocks from
the arbitrary shock force input to a linear shock force which is included in the FRF. The
shock force input may be rewritten as a combination of an arbitrary linear shock forces
and an additional arbitrary force
shocks shocksu Cv F= + ∆ (6.4)
where C is an arbitrary 4x4 diagonal matrix defining the linear shocks. The frequency
response model may now be written as
( ) ( )1
shocks rig rig shocks
rig rig shocks
y I H CS H u H F
y H u H F
−= − + ∆
= + ∆ (6.5)
where S is a matrix that selects the shock velocities from the output vector y. The FRF
matrix rigH now represents the behavior of a damped system with linear shocks defined
by C , while the FRF matrix shocksH represents the influence of any additional damping
force.
The vehicle must be tested using a linearly independent drivefile and for at least two
different shocks at each corner. After all the data is collected for each experiment, F∆ is
calculated at each time step. The data for each setup can be divided into data blocks and
the FFT may be calculated. The measured response for the ith setup and kth data block
may be written as
164
( ) [ ] [ ] ( ) [ ] ( ) [ ]i i irig rig shocksy k H u k H F k kε= + ∆ + (6.6)
The results can be rearranged in matrix form for I shock setups and K data block per
setup as
( ) [ ]
( ) [ ]
( ) [ ]
( ) [ ]
[ ] ( ) [ ]( )
[ ] ( ) [ ]( )
[ ] ( ) [ ]( )
[ ] ( ) [ ]( )
( ) [ ]
( ) [ ]
( ) [ ]
( ) [ ]
11 1
11 1
1 11 1
1 11 1
TTrig
TTrig
Trig
TshocksI ITIT
rig
I ITIT
rig
u Fy
u K F Ky K KH
Hy u F
y K Ku K F K
Y
ε
ε
ε
ε
∆ ∆ = + ∆ ∆
= UH E+
(6.7)
This system of equations along with the error measure in Equation (5.31) yields a least-
squares optimization problem whose solution satisfies the normal equations in Equation
(5.32). This problem will have a unique solution when the input autospectrum matrix is
invertible, which requires the input matrix U to be full-rank. This will tend to be true if
the shocks tested are nonlinear, but the condition number of the input matrix may be large
if the shocks tested are not highly nonlinear. This indicates that the matrix is almost
rank-deficient, which will make the FRF estimates highly sensitive to noise.
One experimental procedure that will encourage good conditioning of the input matrix is
based on the procedure presented in Chapter 5. First, we test with a baseline setup with
near-linear shocks. If we define the diagonal of C to be the slope of the best-fit line for
the force-velocity plot for these shocks, then the F∆ terms will be almost zero for the
baseline setup. Next, we change the left front shock with a significantly different shock
and we repeat the test. This will make the F∆ term for the left front shock large, while
the remaining F∆ terms will be small. Repeating this process for all four corners gives
us an input matrix similar to Equation (5.40), having a significant amount of linearly
independent input energy in each input column.
165
This procedure was used to collect the data for FRF estimation. First, a baseline setup
was tested using the 800s random drivefile with almost linear shocks with a damping
coefficient of 20.74 lb/(in/s). Next, the left front shock was swapped for an alternate
shock with a damping coefficient of 9 lb/(in/s) and the 800s random drivefile was
repeated. This process was repeated at the remaining three corners, where the alternate
shock was run at the corner of interest, while all other shocks were the baseline shocks.
This data gives us the information needed to perform FRF estimation.
6.3.2 FRF Estimation
The data collected from the 800 second drivefile for one baseline setup and four alternate
setups was used to perform FRF estimation. Shock velocity was then calculated from
shock pot data as described above. This shock velocity was used to calculate a linear
shock force with damping coefficient of 20.74 lb/(in/s), which was subtracted from the
measured shock force to obtain the change in shock force defined in Equation (6.4).
Each 800 second data set was then divided into 50 data blocks using a Hamming window
with 50% overlap, providing a frequency resolution of 0.03 Hz. The FFT was calculated
for each data block, and this frequency-domain data was used to form Equation (6.7).
The FRF was then estimated by solving the normal equations defined in Chapter 5. The
total coherence for all outputs was generally high, with a drop in coherence from 4-6 Hz
for the front tire force, which corresponded to an anti-resonance.
To prepare the FRF estimates for transfer function fitting, the FRFs were modified in
regions where the FRF estimate is poor, as discussed in Section 6.2. First, the FRF for
several input-output pair paths were zeroed due to low response levels and poor
coherence. The zeroed paths are denoted by a “0” in Table 6.5. In cases when it was
unclear if the path should be zeroed, the identification and validation process was carried
out both with and without the path present in the mode, and the results were compared to
determine if the path could be zeroed.
166
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency, Hz
Coh
eren
ce
RF Shock VelocityRF Tire ForceFront Ride HeightRear Ride HeightRF Hub Accel
Figure 6.19. Total coherence for select outputs
Table 6.5. Zeroed input-output pairs
LF RF LR RR LF RF LR RR LF RF LR RRLF 0 0 0 0RF 0 0 0 0LR 0 0 0RR 0 0 0LF 0 0 0 0 0RF 0 0 0 0 0LR 0 0 0 0RR 0 0 0 0
Front 0 0Rear 0 0LF 0 0 0 0 0 0 0 0RF 0 0 0 0 0 0 0 0LR 0 0 0 0 0 0 0 0RR 0 0 0 0 0 0 0 0
Out
puts
Shock Velocity
Tire Force
Ride Height
Hub Accels
Wheelloaders Aeroloaders Shocks
Inputs
An important consideration in shock selection is if shock selection at a particular corner
has any influence on sensor response. By zeroing paths for shock inputs in the matrix
shocksH , it is possible that the influence of shock selection at a particular corner on a
particular sensor response may be eliminated. This is more complex than it may seem at
first glance, since changes in shock force input at one corner can also change the shock
167
velocity and shock force at a different corner. For example, while the entries in the FRF
matrix shocksH corresponding to the shock velocity due to left rear shock force are zeroed,
the left rear shock force does affect the shock velocity and shock force at the other three
corners, which do influence the left front shock velocity. Define a logical matrix ijh
which has zeroes if the corresponding entry in shocksH has been zeroed. The number of
nonzero shock velocity feedback paths for the ith output and jth shock force is
12
9ik kj
kh h
=∑ (6.8)
Applying this method for the zeroing described in Table 6.5 yields the matrix shown in
Table 6.6. This shows that despite zeroing some FRF pairs, all shock changes will
influence each output.
Table 6.6. Number of active feedback paths
LF RF LR RRLF 3 2 2 2RF 2 3 2 2LR 2 2 3 2RR 2 2 2 3LF 2 2 2 3RF 2 2 3 2LR 2 2 3 2RR 2 2 2 3
Front 3 3 3 3Rear 3 3 3 3LF 2 2 1 1RF 2 2 1 1LR 1 1 2 2RR 1 1 2 2
Shocks
Inputs
Out
puts
Shock Velocity
Tire Force
Ride Height
Hub Accels
In addition to the input-output pairs that were zeroed because they did not significantly
contribute to the response, the FRF estimate was modified in frequency ranges where it
was poor due to low response level in the given frequency range. Specifically, the FRF
for aeroloader inputs above 5 Hz and for other inputs for the off-diagonal paths at higher
frequencies was replaced with a two-point high frequency residual fit defined in Equation
168
(5.48). This residual fit replaces the poor FRF estimate with a FRF estimate with low
response level, which will aid in fitting transfer functions.
To further smooth the FRF estimate prior to parametric identification, the FRFs are
lowpass filtered. Filtering a complex vector may be performed analogously to filtering a
real vector, with real filters being applied to the magnitude in dB and the unwrapped
phase. A 4th order Butterworth filter was constructed to attenuate FRF variations with a
width of 0.6 Hz or less. This filter was then applied to the magnitude in dB and the
unwrapped phase using the MATLAB function filtfilt to avoid frequency shift. This
eliminates large jumps in the FRF estimates, forcing the independently estimated FRF at
each frequency to behave more smoothly.
The FRF estimates and modified estimates are shown in Figures 6.20-6.23 for shock
velocity, tire force, ride height, and hub acceleration. For brevity, only the results for the
right front and left rear locations are shown for the shock velocity, tire force, and ride
height. The modified FRF estimate is not shown for paths that are zeroed.
0 5 10 15 20-20
-10
0
10
20
30
40
RF
Sho
ck V
eloc
ity /
RF
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz
FRF EstimateModif ied Estimate
0 5 10 15 20-20
-10
0
10
20
30
RF
Sho
ck V
eloc
ity /
LR W
heel
load
er C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-80
-60
-40
-20
0
RF
Sho
ck V
eloc
ity /
RF
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-100
-80
-60
-40
-20
0
RF
Sho
ck V
eloc
ity /
LR A
erol
oade
r Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-70
-60
-50
-40
-30
RF
Sho
ck V
eloc
ity /
RF
Sho
ck F
orce
Cha
nge,
dB
Frequency, Hz0 5 10 15 20
-90
-80
-70
-60
-50
-40
RF
Sho
ck V
eloc
ity /
LR S
hock
For
ce C
hang
e, d
BFrequency, Hz
0 5 10 15 20-30
-20
-10
0
10
20
30
LR S
hock
Vel
ocity
/ R
F W
heel
load
er C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-20
-10
0
10
20
30
40
LR S
hock
Vel
ocity
/ LR
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-200
-150
-100
-50
0
LR S
hock
Vel
ocity
/ R
F A
erol
oade
r Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-60
-50
-40
-30
-20
-10
0
LR S
hock
Vel
ocity
/ LR
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-90
-80
-70
-60
-50
-40
LR S
hock
Vel
ocity
/ R
F S
hock
For
ce C
hang
e, d
B
Frequency, Hz0 5 10 15 20
-50
-45
-40
-35
-30
-25
LR S
hock
Vel
ocity
/ LR
Sho
ck F
orce
Cha
nge,
dB
Frequency, Hz
Figure 6.20. Shock velocity FRF estimate
0 5 10 15 2030
40
50
60
70
80
RF
Whe
ello
ader
For
ce /
RF
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz
FRF EstimateModif ied Estimate
0 5 10 15 2010
20
30
40
50
60
RF
Whe
ello
ader
For
ce /
LR W
heel
load
er C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-60
-40
-20
0
20
40
RF
Whe
ello
ader
For
ce /
RF
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-80
-60
-40
-20
0
20
40
RF
Whe
ello
ader
For
ce /
LR A
erol
oade
r Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-40
-30
-20
-10
0
10
RF
Whe
ello
ader
For
ce /
RF
Sho
ck F
orce
Cha
nge,
dB
Frequency, Hz0 5 10 15 20
-50
-40
-30
-20
-10
0
RF
Whe
ello
ader
For
ce /
LR S
hock
For
ce C
hang
e, d
B
Frequency, Hz
0 5 10 15 2010
20
30
40
50
60
LR W
heel
load
er F
orce
/ R
F W
heel
load
er C
omm
and,
dB
Frequency, Hz0 5 10 15 20
30
40
50
60
70
LR W
heel
load
er F
orce
/ LR
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-40
-20
0
20
40
LR W
heel
load
er F
orce
/ R
F A
erol
oade
r Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-30
-20
-10
0
10
20
30
LR W
heel
load
er F
orce
/ LR
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-70
-60
-50
-40
-30
-20
-10
LR W
heel
load
er F
orce
/ R
F S
hock
For
ce C
hang
e, d
B
Frequency, Hz0 5 10 15 20
-20
-15
-10
-5
0
5
LR W
heel
load
er F
orce
/ LR
Sho
ck F
orce
Cha
nge,
dB
Frequency, Hz
Figure 6.21. Tire force FRF estimate
169
0 5 10 15 20-20
-15
-10
-5
0
Fron
t Rid
e H
eigh
t /
RF
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz
FRF EstimateModified Estimate
0 5 10 15 20-30
-25
-20
-15
-10
-5
Fron
t Rid
e H
eigh
t /
LR W
heel
load
er C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-90
-80
-70
-60
-50
-40
Fron
t Rid
e H
eigh
t /
RF
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-90
-80
-70
-60
-50
-40
Fron
t Rid
e H
eigh
t /
LR A
erol
oade
r Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-140
-120
-100
-80
-60
-40
Fron
t Rid
e H
eigh
t /
RF
Sho
ck F
orce
Cha
nge,
dB
Frequency, Hz0 5 10 15 20
-140
-120
-100
-80
-60
-40
Fron
t Rid
e H
eigh
t /
LR S
hock
For
ce C
hang
e, d
B
Frequency, Hz
0 5 10 15 20-30
-25
-20
-15
-10
-5
Rea
r Rid
e H
eigh
t /
RF
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-20
-15
-10
-5
0
5
Rea
r Rid
e H
eigh
t /
LR W
heel
load
er C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-120
-100
-80
-60
-40
Rea
r Rid
e H
eigh
t /
RF
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-100
-90
-80
-70
-60
-50
-40
Rea
r Rid
e H
eigh
t /
LR A
erol
oade
r Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-140
-120
-100
-80
-60
-40
Rea
r Rid
e H
eigh
t /
RF
Sho
ck F
orce
Cha
nge,
dB
Frequency, Hz0 5 10 15 20
-110
-100
-90
-80
-70
-60
-50
Rea
r Rid
e H
eigh
t /
LR S
hock
For
ce C
hang
e, d
B
Frequency, Hz
Figure 6.22. Ride height FRF estimate
0 5 10 15 20-40
-20
0
20
40
RF
Whe
el A
ccel
/ R
F W
heel
load
er C
omm
and,
dB
Frequency, Hz
FRF EstimateModified Estimate
0 5 10 15 20-60
-40
-20
0
20
RF
Whe
el A
ccel
/ LR
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-120
-100
-80
-60
-40
-20
0R
F W
heel
Acc
el /
RF
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-120
-100
-80
-60
-40
-20
0
RF
Whe
el A
ccel
/ LR
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-100
-90
-80
-70
-60
-50
-40
RF
Whe
el A
ccel
/ R
F S
hock
For
ce C
hang
e, d
B
Frequency, Hz0 5 10 15 20
-120
-100
-80
-60
-40
RF
Whe
el A
ccel
/ LR
Sho
ck F
orce
Cha
nge,
dB
Frequency, Hz
0 5 10 15 20-60
-40
-20
0
20
LR W
heel
Acc
el /
RF
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-40
-20
0
20
40
LR W
heel
Acc
el /
LR W
heel
load
er C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-120
-100
-80
-60
-40
-20
0
LR W
heel
Acc
el /
RF
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-120
-100
-80
-60
-40
-20
0
LR W
heel
Acc
el /
LR A
erol
oade
r Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-110
-100
-90
-80
-70
-60
-50
LR W
heel
Acc
el /
RF
Sho
ck F
orce
Cha
nge,
dB
Frequency, Hz0 5 10 15 20
-120
-100
-80
-60
-40
LR W
heel
Acc
el /
LR S
hock
For
ce C
hang
e, d
B
Frequency, Hz
Figure 6.23. Hub acceleration FRF estimate
6.3.3 Parametric Identification
After performing FRF estimation, the next step is to fit a transfer function to each input-
output pair using invfreqz. The denominator polynomial order was increased from 4 until
acceptable matching was achieved without creating artificial modes. This resulted in
denominator orders ranging from as low as for 4 for aeroloader inputs to as high as 14 for
wheelloader and shock inputs. The weighting functions were
( )
[ )[ )[ )
( ) ( )
[ )[ )[ )
1 0.1,5 1 0.5,100.5 5,7 0.5 10,15
0.1 7,12 0.1 15,200 0
aero wheels shocks
f ff f
w f w f w ff fotherwise otherwise
∈ ∈ ∈ ∈ = = =
∈ ∈
(6.9)
The weighting functions were adjusted slightly as needed to provide better matching in
different frequency ranges. The invfreqz function provided discrete-time transfer
functions with a sample rate of 125 Hz, matching the desired simulation rate. The
170
resulting transfer functions matched the modified FRF estimates well. An example of the
resulting transfer function fits is shown in Figure 6.24.
0 5 10 15 20-20
0
20
40
Frequency, Hz
RF
Sho
ck V
eloc
ity /
RF
Whe
ello
ader
Com
man
d, d
B
FRFTF
0 5 10 15 20-200
-150
-100
-50
0
RF
Sho
ck V
eloc
ity /
LR W
heel
load
er C
omm
and,
dB
Frequency, Hz0 1 2 3 4 5
-46
-44
-42
-40
-38
-36
RF
Sho
ck V
eloc
ity /
RF
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 1 2 3 4 5
-70
-60
-50
-40
RF
Sho
ck V
eloc
ity /
LR A
erol
oade
r Com
man
d, d
B
Frequency, Hz0 5 10 15 20
-50
-45
-40
-35
-30
-25
RF
Sho
ck V
eloc
ity /
RF
Sho
ck F
orce
, dB
Frequency, Hz0 5 10 15 20
-90
-80
-70
-60
-50
-40
RF
Sho
ck V
eloc
ity /
LR S
hock
For
ce, d
B
Frequency, Hz
0 5 10 15 20-200
-150
-100
-50
0
LR S
hock
Vel
ocity
/ R
F W
heel
load
er C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-20
0
20
40
LR S
hock
Vel
ocity
/ LR
Whe
ello
ader
Com
man
d, d
B
Frequency, Hz0 1 2 3 4 5
-80
-70
-60
-50
-40
LR S
hock
Vel
ocity
/ R
F A
erol
oade
r Com
man
d, d
BFrequency, Hz
0 1 2 3 4 5-50
-45
-40
-35
LR S
hock
Vel
ocity
/ LR
Aer
oloa
der C
omm
and,
dB
Frequency, Hz0 5 10 15 20
-90
-80
-70
-60
-50
-40
LR S
hock
Vel
ocity
/ R
F S
hock
For
ce, d
B
Frequency, Hz0 5 10 15 20
-45
-40
-35
-30
-25
LR S
hock
Vel
ocity
/ LR
Sho
ck F
orce
, dB
Frequency, Hz
Figure 6.24. Transfer function fits for shock velocity
6.4 Model Validation
Now that we have identified a vehicle model that predicts the influence of drivefile
selection and shock forces on sensor response, the vehicle model can be combined with
our shock models to provide simulations of 8-post rig tests. First, we describe the
modeling structure created in Simulink. Next, we validate the model accuracy by
comparing simulation and experimental results.
6.4.1 Simulations with Identified Model
The discrete-time transfer function model identified in the previous section can now be
applied to perform simulations to predict the outcome of future 8-post rig tests for
varying drivefiles and shock selections. The transfer function model is converted to
state-space and is then combined with shock models in Simulink, as shown in Figure
6.25.
171
F_linear
u_rig
dF
y
v _shocks
Vehicle Model
Ysim
To Workspace
v _shocks F_shocks
Shock Models
u_rig
Drivefile
K* u
Baseline Shocks
y
v _shocks
F_shocks
dF
Figure 6.25. Simulink model
Simulations were run with a sample rate of 125 Hz, consistent with the sample rate of the
identified discrete-time model. Drivefiles that were run on the 8-post rig were decimated
from the controller frequency of 1250 Hz to 125 Hz using the MATLAB function
decimate, which first lowpass filters the original signal using an 8th order Chebyshev
Type I filter with a cutoff frequency of 80% of the reduced sample rate’s Nyquist
frequency before downsampling the signal. The 8-post rig tests described in Section 6.1
are then reproduced by applying the same drivefiles and using shock models for the same
shock setups. The simulation results are stored for comparison with experimental data
6.4.2 Track-Based Drivefile Testing
The first set of validation results that we show are for the Richmond International
Raceway track drivefile. This is the type of drivefile that race teams would commonly
use during an 8-post rig testing session to characterize their vehicle’s performance on a
particular racetrack as components such as shocks are changed. First, we show the time
response for 4 different signals for the baseline setup in Figure 6.26. As these plots
show, the simulation results show generally good agreement with the experimental data.
The shock velocity shows excellent agreement, which is critical since we will use this
simulated shock velocity to calculate shock force.
172
8 8.5 9 9.5 10-15
-10
-5
0
5
10
time, seconds
LR S
hock
Vel
ocity
, in/
s
DataSim
8 8.5 9 9.5 10-400
-200
0
200
400
600
time, seconds
Tota
l Tire
For
ce, l
bs
8 8.5 9 9.5 10-0.6
-0.4
-0.2
0
0.2
0.4
time, seconds
Fron
t Rid
e H
eigh
t, in
8 8.5 9 9.5 10-2
-1
0
1
2
time, seconds
LR W
heel
Acc
el, g
8 8.5 9 9.5 10-15
-10
-5
0
5
10
time, seconds
LR S
hock
Vel
ocity
, in/
s
DataSim
8 8.5 9 9.5 10-400
-200
0
200
400
600
time, seconds
Tota
l Tire
For
ce, l
bs
8 8.5 9 9.5 10-0.6
-0.4
-0.2
0
0.2
0.4
time, seconds
Fron
t Rid
e H
eigh
t, in
8 8.5 9 9.5 10-2
-1
0
1
2
time, seconds
LR W
heel
Acc
el, g
Figure 6.26. Baseline response for Richmond drivefile
While accurately predicting the time signals is important, it is critical that the model is
able to accurately predict relative trends in the response as the vehicle setup is changed.
RMS-based signal measures are commonly used to characterize the relative behavior of
different setups. Figure 6.27 shows the RMS trending for the signals shown in Figure
6.27 as the shock setup is changed form the baseline setup to the four alternate setups.
The response level for the left rear shock velocity changes the most for a change in left
rear shock, does not change from the baseline significantly for a change in the front
shocks, and only changes slightly for a change in the right rear shock. This trending is
accurately predicted by the model. The simulation trending for the total tire force and
front ride height trend similarly with the data, with the exception of the change in right
front shock not trending as well. The relative trending for the left rear accelerometer
matches extremely well, with the simulation absolute results about 0.4 g higher than the
data.
173
Base Alt LF Alt RF Alt LR Alt RR2
2.5
3
3.5
4
Shock Setup
LR S
hock
Vel
ocity
RM
S, i
n/s
DataSim
Base Alt LF Alt RF Alt LR Alt RR140
150
160
170
180
Shock Setup
Tota
l Tire
For
ce R
MS
, lb
Base Alt LF Alt RF Alt LR Alt RR0.13
0.14
0.15
0.16
0.17
0.18
Shock Setup
Fron
t Rid
e H
eigh
t RM
S, i
n
Base Alt LF Alt RF Alt LR Alt RR
0.35
0.4
0.45
0.5
Shock Setup
LR W
heel
Acc
el R
MS
, g
Figure 6.27. Shock setup RMS trending for Richmond drivefile
The intended purpose of this model is to determine which setups will tend to have a high
RMS level and which will have a low RMS level for various RMS-based measures.
These RMS level predictions can be used to pre-screen potential setups before 8-post rig
testing, eliminating poor setups, selecting good setups, and identifying trends to improve
efficiency of 8-post testing days. The classification of setups as good or bad is based on
the relative RMS response levels, not the absolute RMS levels. If the relationship
between the simulated RMS levels and the measured RMS levels are monotonic,
response levels will be accurately categorized. To clearly see how the simulation and
data trending agree, the RMS response levels for the data is plotted against the simulation
results in Figure 6.28. These plots show that the simulation accurately predicted the trend
except at one point for the total tire force and one point for the front ride height. These
points correspond to the left rear setup.
174
2.2 2.4 2.6 2.8 3 3.22
2.5
3
3.5
4
Sim LR Shock Velocity RMS, in/s
Dat
a LR
Sho
ck V
eloc
ity R
MS
, in/
s
140 150 160 170 180140
150
160
170
180
Sim Total Tire Force RMS, lb
Dat
a To
tal T
ire F
orce
RM
S, l
b
0.144 0.146 0.148 0.15 0.152 0.154
0.14
0.15
0.16
0.17
Sim Front Ride Height RMS, in
Dat
a Fr
ont R
ide
Hei
ght R
MS
, in
0.35 0.4 0.45 0.5
0.35
0.4
0.45
0.5
Sim LR Wheel Accel RMS, g
Dat
a LR
Whe
el A
ccel
RM
S, g
Figure 6.28. Shock setup RMS correlation for Richmond drivefile
6.4.3 Sine and Bump Testing
The next validation is wheelloader sine testing in heave, pitch, and roll on the baseline
setup. This allows us to see how well our linear model can match the gain and phase of
the experimental results. Figure 6.29 shows the sine testing heave time signals for 7
frequencies and 4 signals, while Figure 6.30 shows the corresponding RMS amplitudes.
The simulation results generally look similar to the experimental data, with some phase
or amplitude error. The tire force and wheel accelerometer show a factor of 3 harmonic
below about 6-8 Hz, while the simulation looks sinusoidal. This indicates that the model
does not have sufficient nonlinearity to precisely reproduce the response of these signals
in this frequency range.
175
3 4 5-20
0
20
time, seconds
RF
Sho
ck V
eloc
ity, i
n/s f = 1 Hz
Data Sim Error
14.5 15 15.5-20
0
20
time, seconds
f = 2 Hz
26.8 27 27.2-20
0
20
time, seconds
f = 4 Hz
39.6 39.8-20
0
20
time, seconds
f = 6 Hz
52.4 52.5 52.6-20
0
20
time, seconds
f = 8 Hz
65.3 65.4 65.5-20
0
20
time, seconds
f = 10 Hz
78.2 78.25 78.3-20
0
20
time, seconds
f = 15 Hz
3 4 5
-1
0
1
time, seconds
Fron
t Rid
e H
eigh
t, in
14.5 15 15.5
-1
0
1
time, seconds26.8 27 27.2
-1
0
1
time, seconds39.6 39.8
-1
0
1
time, seconds52.4 52.5 52.6
-1
0
1
time, seconds65.3 65.4 65.5
-1
0
1
time, seconds78.2 78.25 78.3
-1
0
1
time, seconds
3 4 5
-2000
0
2000
time, seconds
Tota
l Tire
For
ce, l
b
14.5 15 15.5
-2000
0
2000
time, seconds26.8 27 27.2
-2000
0
2000
time, seconds39.6 39.8
-2000
0
2000
time, seconds52.4 52.5 52.6
-2000
0
2000
time, seconds65.3 65.4 65.5
-2000
0
2000
time, seconds78.2 78.25 78.3
-2000
0
2000
time, seconds
3 4 5-4
-2
0
2
4
time, seconds
RF
Whe
el A
ccel
, g
14.5 15 15.5-4
-2
0
2
4
time, seconds26.8 27 27.2
-4
-2
0
2
4
time, seconds39.6 39.8
-4
-2
0
2
4
time, seconds52.4 52.5 52.6
-4
-2
0
2
4
time, seconds65.3 65.4 65.5-4
-2
0
2
4
time, seconds78.2 78.25 78.3-4
-2
0
2
4
time, seconds
Figure 6.29. Sine heave time results
1 2 4 6 8 10 150
5
10
15
20
Excitation Frequency, Hz
RF
Sho
ck V
eloc
ity R
MS
, in/
s
DataSim
1 2 4 6 8 10 150
0.5
1
1.5
Excitation Frequency, Hz
Fron
t Rid
e H
eigh
t RM
S, i
n
1 2 4 6 8 10 150
500
1000
1500
2000
Excitation Frequency, Hz
Tota
l Tire
For
ce R
MS
, lb
1 2 4 6 8 10 150
0.5
1
1.5
2
2.5
3
Excitation Frequency, Hz
RF
Whe
el A
ccel
RM
S, g
Figure 6.30. Sine heave testing RMS results
Next, we show the results for bump testing in Figure 6.31 for 4 signals during an upward
wheelloader bump in heave. The data was lowpass filtered with a cutoff frequency of 60
Hz to remove high frequency content. The simulations show good agreement with the
176
measured data for this 0.4 inch peak-peak square wave, indicating that the model
provides a reasonable estimate of vehicle response to short wavelength track inputs
commonly found on the track surface.
5 5.2 5.4 5.6 5.8-10
-5
0
5
10
15
20
time, seconds
RF
Sho
ck V
eloc
ity, i
n/s
DataSim
5 5.2 5.4 5.6 5.8-0.6
-0.4
-0.2
0
0.2
time, seconds
Fron
t Rid
e H
eigh
t, in
5 5.2 5.4 5.6 5.8-1000
0
1000
2000
3000
4000
time, seconds
Tota
l Tire
For
ce, l
b
5 5.2 5.4 5.6 5.8-4
-2
0
2
4
6
time, seconds
RF
Whe
el A
ccel
, g
Figure 6.31. Bump testing results
6.4.4 Static Testing
Static testing was performed by loading and unloading the chassis using a series of 5
static aeroloader forces levels held constant for 5 seconds in heave, pitch, roll, warp, and
single-actuator input configurations. This provides a large dataset to estimate and
validate the static behavior of the vehicle. The average force and position for each static
position was calculated and the data was used to create a least-squares estimate of a
vehicle spring matrix of the form
aero aero zeroF KX F= + (6.10)
where aeroF and aeroX are the force and position of the four aeroloaders, while K and
zeroF are the estimated spring matrix and force at zero position. The spring matrix may
then be transformed if desired using Equations (6.1) and (6.2) to describe the relationship
177
between actuator mode displacements and mode forces. This representation of the
stiffness matrix was found to be
1 3
0.71 0.00 0.11 1.200.14 3.21 0.07 0.57
100.47 0.06 0.66 3.62
1.38 1.92 0.98 27.90
m f xK T KT −
− − − − = = × − − −
(6.11)
where heave force is in lb, heave/pitch/roll moments are in ft-lb, heave displacement is in
inches, and heave/pitch/roll angles are in degrees. The estimated force can then be
compared with the measured force, as shown in Figure 6.32 for the static heave test.
Similar results exist for the other tests.
0 20 40 60 80 100 120
-4
-3
-2
-1
0
time, seconds
LF A
erol
oade
r Pos
ition
, in
DataModel
0 20 40 60 80 100 120
-4
-3
-2
-1
0
time, seconds
RF
Aer
oloa
der P
ositi
on, i
n
0 20 40 60 80 100 120-4
-3
-2
-1
0
time, seconds
LR A
erol
oade
r Pos
ition
, in
0 20 40 60 80 100 120
-3
-2
-1
0
time, seconds
RR
Aer
oloa
der P
ositi
on, i
n
Figure 6.32. Static testing results
The validation results show that the spring model show generally good agreement with
experimental data. The static positions in the left front, right front, and left rear showed
some hysteresis between loading and unloading, possibly due to friction in the suspension
or at the tire-wheel platen interface. A simple spring model cannot account for this
hysteresis, but the weighting on the objective function used in the linear regression could
be adjusted to redistribute the hysteresis error as desired.
If DC response was desired for analysis, it could be added to the simulation. Since we
are investigating how shock selection influences the trends in the vehicle response and
178
the DC actuator inputs do not influence shock velocity, we will not include this in our
model.
6.5 Summary
This section has developed and applied a method to identify a vehicle model using data
collected from an actual 8-post rig test to predict the outcome of future 8-post rig tests for
different drivefile and shock selections. The model was then validated by using it to
perform simulations for several different drivefiles and shock setups and comparing it to
experimental data. This analysis showed us that the model generally does a good job of
predicting the amplitude trends of experiments, with some amplitude and phase error.
There was also nonlinearity not accurately captured by the model in tire force and hub
accelerometer measurements.
179
Chapter 7 Predicting the Influence of Shock Absorber Setup
This chapter applies the vehicle model identified using 8-post rig data in Chapter 6 to
predict the influence of shock absorber setup for a large number of shock setups. After
first describing the shock build database included in the shock selection process, the
results of simulation are shown. The predicted trends are then verified by testing a select
number of setups on the 8-post rig that have been determined to exhibit either a low level
or high level of various RMS-based response measures by the simulation.
7.1 Shock Build Database
To illustrate the ability of our identified model to predict the response levels of various
RMS-based response measures, we must first define the search space of potential shock
setups. The empirical shock model developed in Chapter 3 fits a model to shock
dynamometer data collected from a specific shock build, giving us a model that
accurately represents the performance of a real shock absorber. This shock modeling
structure leads us to use a discrete shock build selection space. If we model N shock
builds and we assume that each shock build can be placed at as many locations as
desired, we have N4 possible shock setups to choose from. Since each simulation of the
Richmond International Raceway takes approximately 30 seconds, an estimate of the
time to run all possible combinations may be calculated, as shown in Table 7.1.
180
Table 7.1. Number of shock builds and simulation times
Shocks Setups Sim Time, h1 1 0.012 16 0.133 81 0.684 256 2.135 625 5.216 1,296 10.807 2,401 20.018 4,096 34.139 6,561 54.6810 10,000 83.33
In practical applications, it will often be desired to optimize a chosen objective function
over a large number of shock builds to find the best setups. While some of the potential
combinations may be eliminated if there objective function shows limited sensitivity to
certain shock selections, it will often be impractical to run all possible combinations.
When the simulation time is too large to consider exhaustive search, more efficient
optimization methods that can deal with discrete search spaces can be considered, such as
genetic algorithms. Since we do not have a desired objective function and our goal is
only to show that our model can distinguish between shock setups that provide low and
high response levels of various response measures, we will choose a smaller shock build
space that will allow us to perform an exhaustive search.
Since we are not trying to cover the entire space of all potential shock builds with our
shock database, our only goal in defining our shock database is to assure that our
different shock builds provide a noticeable change in response to make each simulation
meaningful. To provide good variation between our different shock builds, we first
define low, medium, and high damping levels. Standard shim sets are denoted by letters,
which correspond to a specific shim thickness, as shown in Table 7.2. Each shim set
consists of four shims with the specified stiffness inches and 4 different outer diameters
arranged in a pyramid. In this chapter, we will denote shim settings by the compression
shim stack letter followed by the rebound shim stack letter. For example, a C/B build has
a C shim stack in compression and a B shim stack in rebound.
181
Table 7.2. Shim sizes (adapted from [9], used with permission of Randy Lawrence,
President, Penske Racing Shocks, 2009)
The smallest shim set commonly available is an AA shim set, which consists of four
shims with a thickness of 0.004 inches each. Using this shim set on a linear piston
provides a nominal damping coefficient of approximately 9 lb/(in/s). The alternate shock
used in Chapter 6 used the AA shim stack in both compression and rebound, which we
denote as an AA/AA build. The AA shim stack will be used as our low shim setting.
Next, we define middle and high levels of damping. The B shim stack provides a
nominal damping coefficient of approximately 20 lb/(in/s), while the C shim stack
provides a nominal damping coefficient of approximately 30 lb/(in/s). These shim stacks
provide more than twice and three times the low shim stack damping. These shim stacks
will be used for the middle and high settings. The B shim is used in the baseline shocks
use in Chapter 6 in both compression and rebound, which is denoted as a B/B build.
Now that the low, medium, and high levels of damping have been defined, we will now
select our shock absorbers to be included in our shock database. If we ran all possible
182
combinations of low, medium, and high shim stacks in both compression and rebound,
we would have 9 shocks in our database. Eliminating the combinations low/high and
high/low gives us 7 shocks within our shock database. An exhaustive search with 7
shocks requires 2401 simulations and an estimated simulation time of 20 hours.
Table 7.3. Shock build database
AA B CAA x xB x x xC x x
Rebound Shims
Compression Shims
These 7 shocks were built and tested on the Roehrig shock dynamometer using a drive
profile generated from the left front shock displacement measured during 8-post rig
testing for the Richmond drivefile and the baseline shock setup. Shock models were then
created for each shock build to the shock dynamometer data using the methods described
in Chapter 3. These shock models along with the vehicle model were used to perform all
2401 simulations of every shock setup possible with the 7 shocks using the Richmond
drivefile.
7.2 Simulated Shock Trends
After all 2401 simulations were performed, RMS levels of all signals in each simulation
were calculated for comparison. In this section, we consider the trending for RMS
response levels for the front ride height, total tire force, and right front hub acceleration.
All RMS response levels have been normalized to range from 0 to 1, 0 being the lowest
RMS level seen in the 2401 simulations for that signal, 1 being the highest. The extreme
RMS values for each signal used for normalization are shown in Table 7.4
183
Table 7.4. RMS level extremes
min maxFront Ride Height, in 0.158 0.180Rear Ride Height, in 0.085 0.099RF Wheel Accel, g 0.20 0.28LR Wheel Accel, g 0.32 0.47Total Tire Force, lb 149 204
Figure 7.1 shows the RMS response levels for the front ride height and the associated
shock setups. This figure clearly shows that the best shock setups for front ride height
variation use either a C/C or a C/B shock build. No other front shocks give a front ride
height variation within the lowest 7%. No trends can be noticed for the rear shocks,
indicating that the front ride height has very low sensitivity to rear shock selection.
0 500 1000 1500 2000 25000
0.2
0.4
0.6
0.8
1
Run
Fron
t Rid
e H
eigh
t RM
S
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
LF S
hock
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
RF
Sho
ck
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
LR S
hock
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
RR
Sho
ck
Figure 7.1. Front ride height simulation trending
To see how other response measures can vary given the front ride height RMS level, four
response levels are plotted against front ride height RMS in Figure 7.2. While Figure 7.1
indicated that the rear shock selection had limited influence on front ride height, Figure
7.2 shows that tuning front ride height below 5% forces rear ride height variation to
exceed 30%. If both front and rear ride height were key response measures, a
compromise must be made. There is a similar tradeoff with total tire force – if front ride
184
height RMS is tuned below 2%, the tire force RMS cannot be less than 10%. There is a
strong positive correlation for front ride height and right front hub acceleration,
indicating that both measures will be reduced simultaneously. The left rear hub
acceleration seems to be independent of front ride height, indicating that the two
measures can be tuned independently.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Front Ride Height RMS
Rea
r Rid
e H
eigh
t RM
S
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Front Ride Height RMSR
F W
heel
Acc
el R
MS
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Front Ride Height RMS
LR W
heel
Acc
el R
MS
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Front Ride Height RMS
Tota
l Tire
For
ce R
MS
Figure 7.2. Front ride height simulation tradeoffs
The next response measure that we consider is the RMS total tire force. This shows that
the worst shock builds for the left front shock are the C/C and C/B. These shock builds
were shown to be the best for front ride height. In contrast, these shocks are amongst the
best rear shocks for total tire force.
185
0 500 1000 1500 2000 25000
0.2
0.4
0.6
0.8
1
Run
Tota
l Tire
For
ce R
MS
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
LF S
hock
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
RF
Sho
ck
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
LR S
hock
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
RR
Sho
ck
Figure 7.3. Total tire force simulation trending
Figure 7.4 illustrates the tradeoffs between total tire force and other response measures.
It has already been shown that there is a tradeoff between total tire force and front ride
height. This plot shows us that total tire force, rear ride height, and left rear hub
acceleration can be tuned simultaneously with minimal tradeoff. The front ride height
and right front hub acceleration exhibit a tradeoff with the total tire force.
186
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Total Tire Force RMS
Fron
t Rid
e H
eigh
t RM
S
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Total Tire Force RMS
Rea
r Rid
e H
eigh
t RM
S0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Total Tire Force RMS
RF
Whe
el A
ccel
RM
S
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Total Tire Force RMS
LR W
heel
Acc
el R
MS
Figure 7.4. Total tire force simulation tradeoffs
The last signal that we consider is the right front hub acceleration in Figures 7.5 and 7.6.
The best front shocks for right front hub acceleration are C/C, C/B, and AA/AA. The
rear shocks have a limited influence. It has already been shown that the right front hub
accelerometer is correlated with front ride height and exhibits a tradeoff with the total tire
force. Also shown in Figure 7.6 is that there is a significant tradeoff with rear ride height
and that the two hub accelerations can be tuned independently.
187
0 500 1000 1500 2000 25000
0.2
0.4
0.6
0.8
1
Run
RF
Whe
el A
ccel
RM
S
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
LF S
hock
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
RF
Sho
ck
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
LR S
hock
0 500 1000 1500 2000 2500
AA/AAB/AAAA/B
B/BC/BB/CC/C
Run
RR
Sho
ck
Figure 7.5. Right front hub acceleration simulation trending
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
RF Wheel Accel RMS
Fron
t Rid
e H
eigh
t RM
S
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
RF Wheel Accel RMS
Rea
r Rid
e H
eigh
t RM
S
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
RF Wheel Accel RMS
LR W
heel
Acc
el R
MS
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
RF Wheel Accel RMS
Tota
l Tire
For
ce R
MS
Figure 7.6. Right front hub acceleration simulation tradeoffs
188
7.3 Experimental Results
Twenty shock setups were selected for 8-post rig testing that were predicted to have
either high or low levels of various response measures, as shown in Table 7.5. Four low
setups and four high setups were selected for front ride height RMS. Four low setups and
four high setups were also selected for total tire force RMS. Two low setups and two
high setups were selected for right front hub RMS. These 20 setups were tested on the 8-
post rig using the Richmond drivefile and the experimental results were compared with
the simulation. To compare the ability to predict trends to the run-to-run variability, the
standard deviation of the RMS measures calculated for 5 consecutive runs of the same
Richmond drivefile on the baseline setup. This run-run variation is used to apply error
bars to the comparison plots.
Table 7.5. Shock setups for 8-post rig testing
Low Runs High Runs
LF RF LR RR LF RF LR RR1 C/B C/B AA/B C/C 11 AA/AA AA/AA C/B C/B2 C/B C/B AA/AA C/C 12 AA/AA AA/AA C/B C/C3 C/B C/B AA/AA C/B 13 AA/AA AA/AA C/C C/C4 C/B C/B AA/B C/B 14 AA/AA AA/AA C/C C/B5 B/AA C/B C/B C/B 15 C/B C/B AA/AA AA/AA6 B/AA C/C C/B C/B 16 C/B C/C AA/AA AA/AA7 B/AA C/C C/C C/B 17 C/C C/C AA/AA AA/AA8 B/AA C/B C/C C/B 18 C/C C/B AA/AA AA/AA9 C/B C/B AA/AA B/AA 19 B/AA B/AA C/B C/B10 C/B C/C AA/AA B/AA 20 AA/AA AA/AA C/B C/B
Front Ride
Height
Total Tire
Force
RF Accel
Front Ride
Height
Total Tire
Force
RF Accel
A comparison of the simulated and measured results for front ride height, total tire force,
and right front hub acceleration is shown in Figure 7.7. For each response measure, only
the setups identified for each response measure as a potential high or low setup in Table
7.5 are shown. For the front ride height, the absolute response levels do not match
accurately, but the relative trend does match. This is clearly illustrated when the
simulation and measured front ride height RMS levels are plotted against each other.
Also plotted is a best-fit line for the simulated versus data plot and ±2 standard deviations
from the best fit line. As this plot shows, the data is close to being within the 2 sigma
error bars.
189
1 2 3 4 11 12 13 14
0.14
0.16
0.18
0.2
Setup Number
Fron
t Rid
e H
eigh
t RM
S, i
n
DataSim
0.135 0.14 0.145 0.15 0.155 0.160.12
0.14
0.16
0.18
0.2
0.22
Sim Front Ride Height RMS, in
Dat
a Fr
ont R
ide
Hei
ght R
MS
, in
5 6 7 8 15 16 17 18140
160
180
200
220
Setup Number
Tota
l Tire
For
ce R
MS
, lb
DataSim
140 160 180 200 220160
180
200
220
240
Sim Total Tire Force RMS, lb
Dat
a To
tal T
ire F
orce
RM
S, l
b
9 10 19 200.2
0.22
0.24
0.26
0.28
0.3
Setup Number
RF
Whe
el A
ccel
RM
S, g
DataSim
0.2 0.22 0.24 0.26 0.28 0.3 0.320.2
0.22
0.24
0.26
0.28
0.3
Sim RF Wheel Accel RMS, g
Dat
a R
F W
heel
Acc
el R
MS
, g
Predicted Low Predicted High
Predicted Low Predicted High
Predicted Low Predicted High
Predicted Low Predicted High
Mea
sure
d Lo
wM
easu
red
Hig
h
Predicted Low Predicted High
Mea
sure
d Lo
wM
easu
red
Hig
h
Predicted Low Predicted High
Mea
sure
d Lo
wM
easu
red
Hig
h
1 2 3 4 11 12 13 14
0.14
0.16
0.18
0.2
Setup Number
Fron
t Rid
e H
eigh
t RM
S, i
n
DataSim
0.135 0.14 0.145 0.15 0.155 0.160.12
0.14
0.16
0.18
0.2
0.22
Sim Front Ride Height RMS, in
Dat
a Fr
ont R
ide
Hei
ght R
MS
, in
5 6 7 8 15 16 17 18140
160
180
200
220
Setup Number
Tota
l Tire
For
ce R
MS
, lb
DataSim
140 160 180 200 220160
180
200
220
240
Sim Total Tire Force RMS, lb
Dat
a To
tal T
ire F
orce
RM
S, l
b
9 10 19 200.2
0.22
0.24
0.26
0.28
0.3
Setup Number
RF
Whe
el A
ccel
RM
S, g
DataSim
0.2 0.22 0.24 0.26 0.28 0.3 0.320.2
0.22
0.24
0.26
0.28
0.3
Sim RF Wheel Accel RMS, g
Dat
a R
F W
heel
Acc
el R
MS
, g
Predicted Low Predicted High
Predicted Low Predicted High
Predicted Low Predicted High
Predicted Low Predicted High
Mea
sure
d Lo
wM
easu
red
Hig
h
Predicted Low Predicted High
Mea
sure
d Lo
wM
easu
red
Hig
h
Predicted Low Predicted High
Mea
sure
d Lo
wM
easu
red
Hig
h
Figure 7.7. Comparison of simulated and measured shock trending
For the total tire force, the absolute trends also do not match well, but the relative trends
do match. The data also lies within the ±2 standard deviation error bars.
The right front rear acceleration data appears to match the simulation well, both in
absolute and in relative terms. The data also falls within the 2 sigma error bars.
190
7.4 Summary
This chapter has applied the vehicle and shock models developed in this research to find
the best and worst setups in terms of three different RMS measures. A database of 7
different shock builds was constructed to provide a range of dynamic responses. All
2401 possible shock setups were simulated using the models developed in this research,
and the trends in the RMS response levels were observed for a select group of signals.
Twenty setups were then selected for 8-post rig testing, with each having either a high or
low response level of one of the signals predicted by the simulation. Comparison of the
trends predicted by the simulation and measured on the 8-post rig showed that the
response levels often did not match on an absolute basis, but showed good agreement on
a relative scale. The simulation error was found to be bounded near or within ±2
standard deviations of the baseline response levels.
191
Chapter 8 Conclusions
This chapter summarizes this research and presents conclusions based on that research. It
also describes recommendations for future research that builds on this research or
complementary research that would improve the application of this research.
8.1 Summary and Conclusions
This research has developed a method that uses experimental data to identify a linear
vehicle model that predicts the effect of actuator inputs and shock force on sensor
measurements during an 8-post rig test. This model can be coupled with suspension
component models, such as linear or nonlinear shock absorber models, to simulate how
the vehicle response changes with shock selection. The trends observed in simulation as
components are changed can be used to increase productivity during 8-post rig testing.
Some ways the simulation results may be applied to improve 8-post rig testing efficiency
are: suggesting or eliminating setups to test, identifying tradeoffs in vehicle performance,
identifying sensitivity of setup parameters, and identifying setup parameters which have
limited influence on the response.
This approach differs from previous vehicle modeling studies with component tuning in
that it does not make any assumptions about the physics of the vehicle, other than
assuming the relationship is linear between the shaker rig actuators, shock forces, and the
vehicle sensors.
Chapter 3 developed a physically-motivated nonlinear dynamic shock absorber model
that can be quickly fit to experimental data and implemented in simulation studies. This
model is based on the understanding that the shock is dominantly a velocity-dependent
device, with lag due to compressibility effects. Mechanically, this behavior was shown to
192
be similar to a damper in series and in parallel with a stiffening spring in series. The
effective time constant of such a system can become very small at higher velocities,
making numerical integration very slow. To avoid these numerical efficiency issues, a
slightly different model structure was suggested. The model consists of an algebraic
backbone, which is an algebraic function of velocity, and a nonlinear low-pass filter,
which has been designed based on the observation that shocks often exhibit less
hysteresis at higher velocities. Due to the simplicity of the model, it can be fitted with
data and evaluated quickly, making it ideal for simulation studies.
After fitting several shock models to experimental data, the models were validated using
additional data sets. This analysis showed that the RMS force error in the dynamic
models was 45% to 75% less than the basic polynomial curve fit model, as shown in
Figure 8.1.
Chapter 4 develops the methods needed to develop models of vehicles on the 8-post rig
on a quarter-car scale. By developing our method on this smaller scale, it allowed us to
analyze the method more fully. Chapter 4 develops the vehicle model identification
method at a quarter-car scale. The method was first developed using simulated quarter-
car data with a known model. This allowed comparisons between the ideal model and the
identified model to verify the ID method was accurate. Comparison of the ideal FRF and
the transfer function fit showed perfect FRF matching except at low frequency where the
shock force excitation was low.
After developing the method on simulated quarter-car data, the method was applied to
experimental data collected from a laboratory quarter-car rig. The FRF estimates had
good coherence except where excitation levels were low. The identified model was then
used to perform simulations for different shocks, which was then compared to
experimental results. This analysis showed that the method could predict the quarter-car
rig response trends.
Chapter 5 extends the method to full-vehicle scale in simulation. A seven degree of
freedom vehicle state space model was constructed to generate simulated data sets for the
193
identification process. The identification process can then be judged acceptable if the
identified model is similar to the known ideal model.
The frequency response function (FRF) of the state space model was then explored to
provide insight prior to the identification process. This analysis showed the importance
of both the aeroloaders and the wheelloaders to excite sensor response at low and high
frequencies, respectively.
If the shock force is almost linearly dependent on the wheelloader and aeroloader inputs,
the typical FRF estimation process will fail. A modified FRF estimation process was
described which uses test data from several different shock configurations to address the
shock force dependence issue. This method provided FRF estimates, which matched the
ideal FRF where the coherence was high. When the coherence was poor due to low
shock force or output levels at low or high frequency, the FRF estimate was replaced by a
residual fit with a linear phase and linear dB/decade roll-off. The FRF estimate was
further smoothed by applying a filter, reducing FRF variations with a width of 0.6 Hz or
less.
Once an acceptable FRF estimate was calculated, a parametric model was identified by
fitting transfer functions to each input-output pair. Simulations were run for 5 different
shock configurations and compared to simulation results from the ideal model, showing
that the identified model could accurately reproduce the original response and predict
trends in response metrics.
Chapter 6 develops and applies the vehicle identification method on to data collected
from actual 8-post rig test to predict the outcome of future 8-post rig tests for different
drivefile and shock selections. The coherence for the FRF estimates was generally high
when the response levels were high, with. If the response level was determined to be low
across all frequencies of interest, the corresponding transfer function was zeroed. If the
response level was low over a range of frequencies, the poor FRF estimate was replaced
with a low FRF estimate with linear phase and a linear dB/decade magnitude roll-off
before fitting the transfer function.
194
The model was then validated by performing simulations for several different drivefiles
and shock setups and comparing it to experimental data. This analysis showed us that the
model generally does a good job of predicting the amplitude trends of experiments, with
some amplitude and phase error. There was also nonlinearity not accurately captured by
the model in tire force and hub accelerometer measurements.
Chapter 7 applies the vehicle and shock models developed in this research to find the best
and worst setups in terms of three different RMS measures. A database of 7 different
shock builds was constructed to provide a range of dynamic responses. All 2401 possible
shock setups were simulated using the models developed in this research, and the trends
in the RMS response levels were observed for a select group of signals. Twenty setups
were then selected for 8-post rig testing, with each having either a high or low response
level of one of the signals predicted by the simulation. Comparison of the trends
predicted by the simulation and measured on the 8-post rig showed that the response
levels often did not match on an absolute basis, but showed good agreement on a relative
scale. The simulation error was found to be bounded near or within +/- 2 standard
deviations of the baseline response levels, indicating that the simulated trends match the
measured trends within expected run-to-run variation limits.
8.2 Recommendations for Future Research
Since there has been little research published in the field of dynamic shaker rig testing for
vehicle dynamic applications, this research serves as a starting point. This research has
been a successful first step in identifying dynamic system models from laboratory shaker
rig test data, and it can be used as a starting point for future research. After first
discussing direct improvements and extensions of this research, we will then discuss
complementary research that will improve the value of this research path.
The first suggested improvement of this research is to improve the linear system
identification method used to identify the linear model. This study uses 8-post rig test
data from multiple setups, estimates the FRF, and fits single-input single-output transfer
195
functions to each input-output pair. While this method is easy to implement, it has
several weaknesses.
First, since the transfer function for each input-output pair was fitted independently, this
created problems fitting transfer functions where the coherence was poor. This model
structure has 7-28 transfer function polynomial coefficients for each input-output pair,
allowing it to approximate any FRF desired. This would be good if we always had good
FRF estimates, but this creates issues when coherence is poor. In these cases, there is not
enough information on the true FRF to accurately fit the model. This was addressed in
this study by zeroing transfer functions or replacing the FRF with a low FRF magnitude.
A better approach would be to use a model structure that has less freedom across all
input-output pairs to approximate an arbitrary linear system, and use the reduced degrees
of freedom to fit the model. Multiple input linear system identification methods such as
Eigensystem Realization Algorithm (ERA) and subspace methods identify a linear state
space model with consistent poles, which reduces the sensitivity of modeling error to
poor coherence.
Another interesting observation from this research is that the model was often able to
predict relative trends, but not absolute trends. The measured versus simulated response
level plots often trended along a straight line, indicating the model may be biased by a
scale factor and an offset. In this study, we performed FRF estimation, fitted transfer
functions, and then performed simulation incorporating a nonlinear shock absorber model
in a feedback loop. While this method should provide good results if there were no
modeling error present in these building blocks of the simulation, this process may cause
bias in the model due to the feedback dynamics when modeling error is present. One
way method to improve modeling error is to use an identification method that selects
model parameters to minimize the error between simulation results and data. This will
distribute the error between the FRF building blocks to minimize output error, not try to
minimize error for just one building block at a time. Due to the nonlinear nature of the
shock model, this type of method would require nonlinear identification unless the shocks
were linear.
196
Further consideration should be given to the effect of uncertainties in the input signals.
The H1 FRF estimation method we used minimizes expected modeling error assuming
that the input signals are known and the outputs signals are noisy. In the case of the
wheelloader and aeroloader inputs, this is the case since we define those signals. For the
shock force inputs, however, the measured shock force may have significant noise, which
will bias the FRF estimates. Since the assumption that the input measurements are noise-
free may be poor for shock absorber force inputs, use of methods that account for input
measurement noise, such as H2 or Hα FRF estimation, should be considered.
Another class of model structures that should be considered for several reasons is
physics-based model structures. While these model structures can be made very
complex, a good starting point is the 7 degree of freedom model presented in Chapter 5,
modified to include the increased coupling present in the rear solid-axle suspension. This
model structure will provide a smaller set of parameters compared to generic linear
model structures, making it easier to observe the influence of each parameter on
modeling error. This model structure can also be used for vehicle parameter estimation,
and the estimated parameters can be used in more complex simulations than experienced
in the experimental data, such as full-vehicle simulations.
This research only verified that the model was successful for one specific vehicle within
one vehicle class over a limited sampling of commonly-used drivefiles and shock setups.
Further testing should be conducted to determine under what testing conditions we should
expect this model to be effective and when it may break down. This work should also
provide guidance for developing new methods that are successful when the present
method is not.
Another extension of this research is to accommodate nonlinearities that may be present
in 8-post rig testing, such as progressive spring rates, coil bind and separation, bump
stops, tire separation, and suspension and tire friction. It is possible in theory to address
progressive spring rates, coil bind and separation, and bump stops using the method
developed in this research given the kinematic relationship between shock travel and
spring travel and a component model. In practice this may be difficult due to the high
197
sensitivity to suspension position. It is also theoretically possible to apply our
substructuring approach to address tire separation, but this will be difficult on the 8-post
rig since there are 6 forces and moments acting as inputs to the tire contact patch that
may influence sensor response without the sufficient richness of inputs required to
perform identification. Suspension friction should be minimized for good performance,
and tire friction should be minimized during the 8-post test to improve testing
consistency.
One limitation of our shock setup selection method was that we were optimizing the
shock setup over a discrete set of shock models. The advantage of this approach is that
all shock models represent real shocks that can actually be put on a car. The first
disadvantage is that all potential shocks must be built, dynamometer tested, and modeled
before selection can begin. This may not be a huge disadvantage for race teams that have
a large database of shocks data, but it does limit exploring new shock builds. Second,
optimizing over a discrete search space is much more computationally expensive than
searching over a continuous search space. Research to develop a continuous space of
shock models should be considered for various applications. This would allow searching
over a larger search space more efficiently. While the results of the optimization over a
continuous shock database may not provide a setup consisting of four known shock
builds, it will provide a target shock force-velocity curve, which could be used to aid
shock building and design.
Some options for continuous shock parameterizations useful for optimization include
1. Physics-based shock models,
2. Interpolation of shock models for existing builds, and
3. Shape functions.
Physics-based models are useful since their parameters reflect physical properties that
can be used to design shock internals, and the limitations in parameter values lead to
practical limitations in shock behavior, but they are often computationally inefficient. If
there is a mesh of shock models for existing builds, it may be possible to interpolate to
provide an estimate of shock performance within the mesh. For example, we modeled an
198
AA/AA and a B/B build, which have a shim thickness of 0.004 and 0.008 inches
respectively. Without further data, it would seem reasonable to approximate the shock
force from a 0.006 inch A/A shock build as an average of the AA/AA and B/B shock
model force. This type of model would provide the advantages of continuous
parameterization, physical significance of each model, and numerical efficiency. The
main limitation of this approach is the requirements for a mesh of shock model wide
enough to find the best shocks but fine enough to accurately model performance. To
explore trends with no data requirements, a shape function model is ideal. A shape
function model is an algebraic function whose parameters control specific portions of the
curve. Using a shape function model can be useful to explore trends, but the curves often
do not reflect physical constraints of real shock absorbers, so they may lead to results that
are not achievable with real shocks if the model structure and parameters are not properly
constrained.
One critical aspect of 8-post rig testing that was not addressed in this research is how to
apply 8-post rig testing to improve vehicle dynamics on the track. Several studies to
determine how to apply rig data to tune track performance are needed. As discussed in
Chapter 2, since the dynamics on the 8-post rig and on the track are not consistent,
assuming that the sensor measurements on the 8-post rig and on the track will be the
same may lead to erroneous conclusions. A study should be conducted to determine to
what extent and under what conditions trends in rig sensor measurements are correlated
with track measurements. Different 8-post rig drivefile generation methods should be
used, which may include drivefile iteration for a baseline setup, drivefile iteration for
multiple setups, terrain profile-based wheelloader excitation, and aerodynamic model-
based aeroloader force modification.
Also since yaw dynamics are not significant on the rig, it is challenging to optimize the
true metric of race car performance – lap time. Work should be done to integrate the
vehicle behavior observed on the 8-post rig to the additional dynamics observed on the 8-
post rig to determine how changes on the 8-post rig influence track performance. This
can be done using the 8-post rig as a hardware-in-the-loop test with the remaining
199
dynamics running on a real-time processor, or data collected from the 8-post can be used
to perform system or parameter identification for use in simulation.
200
References
1. Kelly, J. Kowalczyk, H. Oral, H. Track Simulation and Vehicle Characterization
with 7 Post Testing. SAE Technical Papers 2002-01-3307.
2. Menzer, J. Seven-post shakers stir up Cup Series' team hierarchy.
NASCAR.com, June 2007.
3. Craig, R. Blades, E. Substructure System Identification: Reduced-Order Models.
Dynamics and Control of Space Structures III, pp 365-379, 1996.
4. Su, T. Juang, J. Substructure System Identification and Synthesis. Journal of
Guidance, Control, and Dynamics, vol 17, no 5, 1994.
5. Lee, D. Hwang, W. Parametric Optimization of Complex Systems Using a Multi-
Domain FRF-Based Substructuring Method. Computers and Structures, vol 81,
pp 2249-2257, 2003.
6. Sjovall, P. Abrahamsson, T. Substructure System Identification from Coupled
System Test Data. Mechanical Systems and Signal Processing, vol 22, pp 15-33,
2008.
7. Kasprzak, J. Floyd, R. Use of Simulation to Tune Race Car Dampers. SAE
Technical Papers 942504.
8. Warner, B. Rakheja, S. An Investigation of the Influence of High Performance
Dampers on the Suspension Performance of a Quarter Vehicle. SAE Technical
Papers 962552.
9. Penske Racing Shocks 7000 Series Technical Manual, 2007.
10. Lang, H. Automotive Dampers at High Stroking Frequency. Ph.D. Thesis,
University of Michigan, 1977.
201
11. Reybrouck, K. A Non Linear Parametric Model of an Automotive Shock
Absorber. Vehicle Suspension System Advancements, SAE SP-1031, February
1994.
12. Talbott, M. An Experimentally Validated Physical Model of a High-Performance
Automotive Monotube Damper. Master's Thesis, Purdue University, 2002.
13. Emmons, S. Boggs, C. Ahmadian, M. Parametric Modeling of a Highly-
Adjustable Race Damper. ASME International Mechanical Engineering Congress
and Exposition, 2006.
14. Rao, M. Gruenberg, S. Torab, H. Measurement of Dynamic Properties of Shock
Absorbers for NVH. SAE Technical Papers 1999-01-1840.
15. Kowalski, D. Rao, M. Blouh, J. Gruenberg, S. Dynamic Testing of Shock
Absorbers under Non-Sinusoidal Conditions. Journal of Automobile Engineering,
vol 216, pp 373-384, 2002.
16. Cafferty, S. Characterization of Automotive Dampers using Higher Order
Frequency Response Functions. Journal of Automobile Engineering, vol 211, pp
181-203, 1997.
17. Yung, V. Wavelet Analysis of High-Frequency Damper Behavior. Journal of
Automobile Engineering, vol 219, pp 977-988, 2005.
18. Kasteel, R. Cheng-guo, W. Lixin, Q. Jin-zhao, L. Wen-zhang, Z. A New Shock
Absorber Model with an Application in Vehicle Dynamic Studies. SAE
Technical Papers 2003-01-3411.
19. Zhou, W. Chew, C. Hong, G. Inverse Dynamics Control for Series Damper
Actuator Based on MR Fluid Damper. IEEE/ASME International Conference on
Advanced Intelligent Mechatronics, 2005.
20. Barber, A. Accurate Models for Complex Vehicle Components using Empirical
Methods. SAE Technical Papers 2000-01-1625.
202
21. Leser, C. Renner, T. Salmon, D. Accurate Shock Absorber load Modeling in an
All Terrain Vehicle Using Black Box Neural Network Techniques. SAE
Technical Papers 2002-01-0581.
22. Savaresi, S. Silani, E. Montiglio, M. Stefanini, A. Previdi, F. A Comparative
Analysis of High-Accuracy Black-Box and Grey-Box Models of MR-Dampers
for Vehicle Control. SAE Technical Papers 2004-01-2066.
23. Miller, A. Testing a Formula SAE Racecar on a Seven-Poster Vehicle Dynamics
Simulator. SAE Technical Papers 2002-01-3309.
24. Kowalczyk, H. Damper Tuning with the use of a Seven Post Shaker Rig. SAE
Technical Papers 2002-01-0804.
25. Chan, B. Sandu, C. Development of a Virtual 7-Post Test Rig for Experimental
Validation. ASME International Mechanical Engineering Congress and
Exposition, 2006.
26. Vilela, D. Vehicle Dynamics Simulation Correlation with Field Maneuvers. SAE
Technical Papers 2001-01-3799.
27. Dittman, K. Albright, F. Leser, C. Validation of Virtual Prototypes via a Virtual
Test Laboratory. MTS White Paper.
28. Hu, H. Experimental Validation of a Half-Vehicle Suspension Model. SAE
Technical Papers 931966.
29. Ballard, R. Overview of Emerging Technology for Laboratory Based
Measurement of Vehicle Performance. SAE Technical Papers 921068.
30. Cheli, F. Sabbioni, E. A Dynamic Light-Duty Vehicle Model: Validation with
Indoor and Outdoor Experimental Tests. Proceedings of the ASME 2007
IDETC/CIE 2007 DETC2007-34753.
203
31. Andersen, Erik. Multibody Dynamics Modeling and System Identification for a
Quarter-Car Test Rig with McPherson Strut Suspension. Master's Thesis,
Virginia Tech, May 2007.
32. Ziegenmeyer, J. Estimation of Disturbance Inputs to a Tire Coupled Quarter-Car
Suspension Test Rig. Master's Thesis, Virginia Tech, May 2007.
33. Alasty, A. Ramezani, A. Genetic Algorithm-Based Parameter Identification of a
Nonlinear Full-Vehicle Ride Model. SAE Technical Papers 2002-01-1583.
34. Juang, J. Applied System Identification. Prentice Hall, 1993.
35. Roehrig Engineering. 2K Electro-Magnetic Actuator Product Description.
www.roehrig.com
36. Boggs, C. Borg, L. Ostanek, J. Ahmadian, M. Efficient Test Procedures for
Characterizing MR Dampers. ASME International Mechanical Engineering
Congress and Exposition, 2006.
37. Oppenheim, A. Discrete-Time Signal Processing. Prentice Hall, 1999.
38. Soderstrom, T. Ljung, L. Gustavsson, I. Identifiablilty Conditions for Linear
Multivariable Systems Operating Under Feedback. IEEE Transactions on
Automatic Control, vol 21, no 6, pp 837-840.
39. Ljung, L. System Identification: Theory for the User. Prentice Hall, 1999.
40. MATLAB R2006b Documentation.